Non-coherent sequence estimation receiver for modulations

ABSTRACT

It is described a non-coherent receiver for digital modulated radio signals, linear or not, possibly affected by channel coding. In the case of M-PSK, or M-QAM modulations, possibly combined with a channel coding, the signal received is demodulated and filtered by a filter matched to the pulse transmitted. In the case of linear modulations in presence of ISI, and of non-linear M-CPM modulations, the reception filter is a matched and whitened one. The signal is filtered and sampled and the samples are accumulated in a relevant phase reconstruction memory containing N−1 samples preceding the actual one. Samples are processed making use of a Viterbi processor estimating the sequence of symbols transmitted according to maximum likelihood. The first embodiment calculates the branch metrics using an estimation of the carrier phase in PSP mode (Per-Survivor-Processing). This improves the performance of the receiver. In the second embodiment, the trellis branch metrics are obtained from the expression for the maximun likelihood sequence estimation. The Viterbi algorithm can be applied in this case.

FIELD OF THE INVENTION

[0001] The present invention relates to the field of digital modulated radio signals and more in particular to a method for the implementation of a non coherent sequence estimation receiver for linear digital modulations. The communication channel is assumed at the beginning as an ideal one, affected by additive white Gaussian noise (Additive White Gaussian Noise, or AWGN). Afterwards, the channel ideality assumption is removed and the presence of intersymbol interference (ISI; InterSymbol Interference) is considered in the demodulated signal. In the considered technique field, there are different classes of receivers for such channels.

BACKGROUND ART

[0002] A first class of receivers is based on the structure of an optimum coherent receiver, that is, a receiver that minimizes the error possibility on symbols decided should the synchronism be perfectly known, and in particular, the phase of the signal received, which will be dealt with hereafter. The implementation of such a receiver does not show particular problems in a laboratory environment, where in fact, the modulation carrier is always available, but it cannot be followed up in practice when this receiver is placed in field and the carrier is not available. In these cases, a preferred solution is to supply the receiver with a synchronization device enabling to <<recover>> the information on the phase of the modulated carrier. The devices more used to this purpose are phase locked loops (Phase Locked Loop, or PLL). Such a receiver shall be hereinafter defined <<pseudocoherent”, since it is implemented according to the configuration of a coherent receiver to which a phase reference is supplied by said synchronization device. In these receivers the phase is recovered at less than 2π/n multiples, where n depends on the type of modulation adopted. As a consequence of the ambiguity on the phase introduced by the PLL, a differential coding must be used in transmission, that is a coding where the information is not associated to the absolute phase of the modulation carrier, but to the phase difference between two consecutive symbols. As an alternative to the differential coding it is possible to use pilot symbols during transmission, as described hereafter.

[0003] A second class of receivers consists of non-coherent receivers, that is those not requiring the information on the absolute phase of the transmitted signal. These receivers have different advantages compared to pseudo coherent receivers, namely:

[0004] 1. They can be employed in situations where the synchronization recovery results to be difficult, such as for instance in the case of fading channels, or in presence of shift Doppier, or of frequency jumps due to the instability of oscillators;

[0005] 2. They are simpler and cost effective since they have no PLL;

[0006] 3. The synchronization state is not lost, contrarily to receivers with PLL where this loss can occur due to phase jumps, false locking or loss of the locking state;

[0007] 4. after an out-of-duty interval caused by deep fading they are immediately operative, contrarily to receivers with PLL that require a transient period to recover the locking condition;

[0008] 5. they can be employed in time division multiple access communication systems (Time Division Multiple Access, or TDMA), where the coherent detection is not recommended due to the comparatively long acquisition time of the synchronism.

[0009] The first non coherent receivers considered in technical literature were differential receivers, often employed in the detection of modulated phase digital signals, or PSK (Phase Shift Keying), where a differential coding ties the information to the phase difference between two consecutive PSK symbols. The receiver estimates this phase difference, not requiring therefore to be locked in phase with the signal received. A possible interpretation of the operation of these receivers is the following: with the differential coding process, the phase reference necessary for the data estimate is contained in the preceding symbol. Therefore it is not necessary to determine an absolute phase reference, since the preceding symbol can be used to this purpose. However, this involves a degrade of performance compared to a coherent receiver, due to the fact that in differential detection the phase reference is noisy, while in the coherent detection this reference is perfectly known and therefore noise free. We could say that in the case of differential detection the signal to noise ratio (Signal-to-Noise Ratio, or SNR) of the reference signal is the same of the SNR of the information signal. In the case of a coherent receiver, on the contrary, the SNR of the reference signal is infinite from the theoretical point of view. For instance, in the case of PSK modulations with two phase values only, or BPSK, (Binary PSK) the loss is small, that is 0,8 dB approximately at bit error rate, or BER, (Bit Error Rate) of 10⁻⁵. On the contrary, in the case of PSK modulations with M>2 phase values, or M-PSK, the performance loss can reach 3 dB.

[0010] Starting from the above considerations, differential receivers have been conceived, drawing the phase reference from a given number of symbols passed, in order to <<filter” the noise effect. In this way the SNR of the phase reference changes to a higher quality and the performance come closer to those of a coherent receiver. This type of receivers employing a so-called <<decision reaction>> are described, for instance in the following papers:

[0011] <<The phase of a vector perturbed by Gaussian noise and differentially coherent receivers>>, authors: H. Leib, S. Pasupathy, published on IEEE Trans. Inform. Theory, vol. 34, pp.1491-1501, November 1988.

[0012] <<Bit error rate of binary and quaternary DPSK signals with multiple differential feedback detection>>, author F. Edbauer, published on IEEE Trans. Commun., vol. 40, pp. 457-460, March 1992.

[0013] They can be considered the forerunners of block differential receivers, or N-differential, described below.

[0014] Block differential receivers fill the performance gap between coherent performance and simply differential ones, and are well described in the following papers:

[0015] <<Multi-symbol detection of M-DPSK”, authors: G. Wilson, J. Freebersyser and C. Marshall, published following the Proceedings of IEEE GLOBECOM, pp.1692-1697, November 1989;

[0016] <<Multiple-symbol differential detection of MPSK>>, authors: D. Divsalar and M. K. Simon, published on IEEE Trans. Commun., vol. 38, pp.300-308, March 1990;

[0017] <<Non-coherent block demodulation of PSK”, authors: H. Leib, S. Pasupathy, published following the Proceedings of IEEE VTC, pp.407-411, May 1990;

[0018] and in the volume under the title <<Digital communication techniques>>, authors: M. K. Simon, S. M. Hinedi e W. C. Lindsey, published by Prentice Hall, Englewood Cliffs, 1995, for the case of M-PSK modulations.

[0019] Block differential receivers, as well as those adopting the <<decision reaction>>, are grounded on the idea to extend the observation interval on which decisions are based, compared to the observation interval of two symbols only, typical of simply differential receivers. For these last, there is an additional peculiarity, which is that to decide on more symbols at the same time, instead of symbol by symbol. N-differential receivers use an observation window of N symbols, and simultaneously make the decision on N−1 information symbols. This decision strategy can be seen as an extension of the decision strategy of differential receivers, which in fact correspond to case N=2. It has been demonstrated that in the case of M-PSK modulations, for N→+∞ the performance of this type of receivers tend to those of the coherent receiver. Number of examples of block differential receivers can be found in literature, suitable to the different modulations; some of them are described in the papers mentioned above. In addition, we point out that:

[0020] M-PSK modulations with channel coding are described in the paper under the title <<The performance of trellis-coded MDPSK with multiple symbol detection”, authors: D. Divsalar, M. K. Simon e M. Shahshahani, published on IEEE Trans. Commun., vol. 38, pp.1391-1403, September 1990;

[0021] M-QAM coded and uncoded modulations (Quadrant Amplitude Modulation) are dealt with in the essay <<Maximum-likelihood differential detection of uncoded and trellis coded amplitude phase modulation over AWGN and fading channels—metrics and performance”, authors: D. Divsalar e M. K. Simon, published on IEEE Trans. Commun., vol. 42, pp.76-89, January 1994;

[0022] M-PSK and M-QAM modulations in fading channels, are treated in the previous article and in the one under the title <<Optimal decoding of coded PSK and QAM signals in correlated fast fading channels and AWGN: a combined envelope, multiple differential and coherent detection approach”, authors: D. Makrakis, P. T. Mathiopoulos and D. P. Bouras, published on IEEE Trans. Commun., vol. 42, pp.63-75, January 1994.

[0023] Some disadvantage, common to all the block differential or N-differential receivers, described in the extensive literature mentioned above, are caused by the type of strategy used in the decision, consisting in an exhaustive research made on the single data blocks. Therefore it is necessary to use small N values, otherwise calculations would exceedingly complicate even for small sizes of the input alphabet, practically impairing the realization of the receivers. To overcome this difficulty, the skilled in the art could think to estimate the sequence transmitted using the Viterbi algorithm, however he should come shortly to the conclusion that this way is not practicable since the metric can be made recurrent in no one of the receivers described. In the light of the above, some N-differential receivers are known, employing, though inappropriately, the Viterbi algorithm. In the case of M-PSK modulations, these receivers have been, described in the following articles:

[0024] <<Non-coherent coded modulation”, author D. Raphaeli, published on IEEE Trans. Commun., vol. 44, pp.172-183, February 1996;

[0025] <<A Viterbi-type algorithm for efficient estimation of M-PSK sequences over the Gaussinn channel withi unknown carrier phase”, authors: P. Y. Kam and P. Sinha, published on IEEE Trans. Commun., vol. 43, pp.2429-2433, September 1995.

[0026] Non coherent receivers described by D. Raphaeli, representing the more pertinent known art, are based on maximally overlapped observations (maximally overlapped observations), that is extended to N−1 symbols preceding the present one, assumed as independent, even if they are not in reality, as the author clearly admits. We can also observe that metrics used are identical to those euristically assigned to the most recent symbols in the receivers described by P. Y. Kam and P. Sinha, where decisions are locally made, at each node of a trellis diagram. In this case there is no accumulation of metrics as, on the contrary, it occurs in the classical Viterbi algorithm. The interesting thing to be noticed in receivers described by D. Raphaeli is that they reach a good operation performance, though inappropriately using the Viterbi algorithm. The approximation introduced, consists in having in a recurrent way, the metrics of the previous N-differential block receivers to the sole purpose of employing the Viterbi algorithm, but without basing the assumptions on metrics and their use in the algorithm context, on effective and convincing theoretical postulates, justifying this recurrence relation. The performance of these receives, while good, find however a limit in the approximation introduced.

OBJECT OF THE INVENTION

[0027] Therefore scope of the present invention is to additionally improve the performance of the known non-coherent receivers, at equal complexity level, or to reduce the complexity, at equal performance, and to indicate a non coherent reception procedure of coded symbol sequences transmitted on a communication channel, affected by additive white gaussian noise, based on a more effective use of the Viterbi algorithm for the maximum likelihood estimation of the sequence transmitted. A receiver is also indicated, performing the above mentioned procedure.

SUMMARY OF THE INVENTION

[0028] To attain these objects, scope of the present invention is a non coherent reception procedure of coded symbol sequences, obtained by amplitude and/or phase digital modulation of a carrier, transmitted on a communication channel, affected by additive white gaussian noise, based on the use of the Viterbi algorithm applied to a trellis sequence diagram, or trellis, where the branches represent all possible transitions among states defined by all possible subsequences of information symbols, possibly coded, of finite length, through which algorithm paths are selected at each symbol interval on the trellis such that a path metric, comulative of transition metrics, is higher, said path metric being an indication of the likelihood level existing between the symbols of a path associated to the same and a transmitted sequence of symbols, characterized in that each said transition metric is calculated through the following steps:

[0029] a) non coherent conversion in base band of the signal received, subsequent filtering of the signal converted through a filter matched to the basic pulse of the received signal, frequency sampling of the symbol of filtered signal, obtaining complex sequential samples;

[0030] b) construction of a phase reference through accumulation of N−1 produced among said complex sequential samples, conjugate, and corresponding coded symbols, also complex, univocally associated to a said relevant branch of the trellis; the number N−1 being said finite length, selected in order to obtain the desired accuracy in the constructed phase, said accuracy increasing as N increases, without excessively increasing the complexity of the trellis, expressed in terms of number of states;

[0031] c) normalization of the value of said phase reference, through division by the modulus of the same;

[0032] d) Replacement of a phase reference, or phasor, of said modulated carrier, present in the known analytical expression of the transition metrics used by an optimum coherent receiver, which could replace said non coherent receiver whenever said phasor is known, with said phase reference constructed to the previous steps, obtaining an analytical expression for the calculation of each said transition metric used by said non coherent receiver, as described in the independent claim 1.

[0033] In a second embodiment of the invention, the expression for the calculation of the trellis branch metrics is obtained from the expression known for the maximum likelihood sequence estimation used by the non-coherent receiver. To this purpose, it is interpreted the function to maximize like a sequence general metric, which can be obtained updating in a recurrent way a sequence partial metric defined at the n-th signal interval, this last being in its turn possible to be calculated through accumulation of incremental metrics of unlimited memory. A truncation at N−1 symbols preceding the present one in the calculation of incremental metrics enables, without a significant information loss increase, the construction of a trellis to which the Viterbi algorithm can be applied for the research of the maximum value path metric, according to the known method, as described also in the independent claim 7.

[0034] A non-coherent receiver realized irrespectively of one or the other embodiments of the present invention, is suitable to process both linear modulated signals, also affected by intersymbol interference, and OPM linear modulated ones, always giving performance higher than all the conventional non-coherent receivers. A single invention concept lays between the two embodiments of the invention, leading to obtain for linear modulations a common general diagram of the receiver, and in the case of M-PSK modulations, also the same analytical expressions of the branch metrics.

[0035] Like the conventional N-differential receivers, also the receiver implemented according to the present invention contains a phase reconstruction memory, or a comparable one, whose length N can be selected in order to obtain a satisfactory compromise between complexity and performance. In fact, as N increases, the performance comes closer to that of the best coherent receiver (which perfectly knows the synchronism and can be implemented in practice only in an approximate way through a pseudocoherent receiver), but at the same time increases also the complexity expressed by the number of states of the trellis diagram. However, it is possible to obtain, with not too high N values, a small complexity and performance very close to the best. When the phase reconstruction memory of the subject receiver assumes a value equal to the length of a block in the N-differential receiver of the known art, or to the observation interval in the receiver described in the essay of D. Raphaeli of February 1996, the considered receiver shows a performance, because it employs an expression more effective of the branch metric, that is better compatible with he subsequent processing steps of the Viterbi algorithm and with the theoretical assumption on which said algorithm bases.

[0036] An additional scope of the invention is a maximum likelihood sequence estimation receiver for coded symbols, implementing the procedure forming the object of claim 1, as described in the independent claim 22.

[0037] Another scope of the invention is a variant of the previous receiver valid when coded symbols relate to a phase numeric modulation, or M-PSK, as described in the independent claim 24.

BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Additional scopes and advantages of the present invention will result more clear from the following detailed description of an implementation of the same and from the attached drawings given as an example but not limited to the same, where:

[0039]FIG. 1 shows an equivalent model in base band of a generic digital communication syster-including a RIC receiver implementing the process of the present invention;

[0040]FIG. 2 shows a block diagram valid to describe the operation of the RIC receiver of FIG. 1 in the case of linear modulations;

[0041]FIG. 3 shows a block diagram valid to describe the operation of the METRICTOT block of FIG. 2;

[0042]FIG. 4 shows a block diagram valid to describe the operation of a METRIC(s) generic block of FIG. 3;

[0043]FIG. 5 shows the implementation of the coding block COD of FIG. 1, in case it represents a convolutional coder for muitiiayer symbols of the MI-PSK type;

[0044]FIGS. 6 and 7 represent two different implementations of the coding block COD of FIG. 1 for two different code typologies, and

[0045]FIG. 8 shows a block diagram valid to describe the operation of the RIC receiver of FIG. 1, in case of non linear modulations of the CPM (Continuous Phase Modulation) type.

[0046] Making reference to FIG. 1, it can be noticed a transmitter TRAS connected to a receiver RIC through a communication channel CAN, that in the more general case dealt with in the present invention, is considered linear, dispersive, and affected by additive white gaussian noise (Additive White Gaussian Noise, or AWGN), with power spectrum density N_(o)/2; in less general cases, the channel is considered as ideal. In the figure, the channel CAN is modelled by the cascade of a DISP block, of a multiplier 1, and of an adder 2. The input of the transmitter TRAS is reached with 1/T time interval, digital information symbols, belonging to a cardinal alphabet M′, assumed to be equipossible and independent, forming a sequence a={a_(n)} reaching the input of a coder COD. Through some coding rule, this last generates at output a sequence of coded symbols, c={c_(n)}, complex in general and belonging to an alphabet with M>M′ cardinality. Notice that being the block diagram of FIG. 1 but an equivalent in base band of the communication system, the signals appearing are actually complex envelopes. The coded sequence {c_(n)} reaches the input of a linear modulator MOD, mapping the sequence in a continuous time signal s(t, a). This signal shall depend of course on the sequence of information symbols briefly indicated by vector a. The transmission signal s(t, a) coming out from the modulator MOD crosses the transmission means used by the communication channel CAN and reaches a receiver RIC implemented according to the invention. It has at its input a signal received r(t), and gives at output an estimate {â_(n)} of the transmitted information sequence {a_(n)}. During the channel crossing, the pulses of the transmission signal s(t, a) undergo, in general, a distortion,and a global phase rotation θ. The DISP block is equivalent to a filter, introduced in the diagram to consider the time dispersion underwent by pulses, while the multiplier 1, to the second input of which a phasor e^(jθ) is applied, considers the above mentioned phase rotation. Finally, the adder 2 adds to the transmission signal coming out from the multiplier 1 the complex envelope w(t) of the noise present on the channel.

[0047] For what said above, the signal received r(t) assumes the following expression:

r(t)=s′(t,a)e ^(jθ) +w(t)  (0.1)

[0048] where the phase rotation θ is assumed as constant for the whole transmission time and modelled as uncertain variable with uniform distribution in the interval [0,2,π] and s′(t, a) denotes the respons of filter DISP to signal s(t, a).

[0049] The schematisation of FIG. 1 is purposely generic, since its sole purpose is to introduce the basic elements of the channel upstream the receiver RIC, where the invention actually resides. Time by time, according to the particular receiver considered, the typologies of COD and MOD blocks shall be specified, as well as the actual characteristics of the channel CAN. Without detriment to the general character of the invention, we shall firstly describe a first embodiment of the receiver for linear modulations M-PSK and M-QAM of signals transmitted on an ideal channel, that is without DISP block. Afterwards, we shall depict a second embodiment of the receiver for linear modulations M-PSK and M-QAM of signals transmitted on a channel alternatively considered as dispersive, and not, and finally for non-linear modulations of the CPM type. In both the embodiments of the invention, the structure of the RIC shown in FIG. 2 is valid, limited to linear modulations; of course, the content of some blocks will change. In the figure we have neglected the blocks not considered absolutely necessary to the understanding of the operation, known to the skilled in the art. FIG. 8 shows the structure of the receiver RIC valid for the CPM modulation, dealt with only for the second embodiment of the invention.

[0050] Making reference to FIG. 2, we shall now describe the receiver RIC valid for the first embodiment of the invention in presence of linear modulations and channel supposed as ideal. In this case the signal s(t,a) will assume the following expression: $\begin{matrix} {{s\left( {t,a} \right)} = {\sum\limits_{i}{c_{i}{h\left( {t - {iT}} \right)}}}} & (0.2) \end{matrix}$

[0051] where T is the symbol interval and h(t) is the pulse transmitted, duly normalized.

[0052] In the assumption to perfectly know the symbol synchronism and the carrier frequency, the signal r(t) can then be expressed as follows: $\begin{matrix} {{{r(t)} = {\sum\limits_{i}c}},{{{h\left( {t - {iT}} \right)}^{j\quad 9}} + {w(t)}}} & (1) \end{matrix}$

[0053] This signal reaches the input of a reception filter FRIC, downstream which a sampler CAMP is placed, withdrawing the samples x_(n) with cadence equal to the symbol frequency 1/T. The samples x_(n) form a sequence {x_(n)} sent to a chain of N−1 delay elements τ₁,τ₂, . . . , τ_(N−1) of a symbol interval T. These elements τ are the flip-flops of a scroll register SHF1 storing a string of N−1 said samples of the filtered signal for the duration of a symbol, making them at the same time available at the output of each single flip-flop τ. The sample x_(n) and the N−1 preceding samples x_(n−1), x_(n−2), . . . , x_(n−N+1) are sent to a METRICTOT block performing, in correspondence, the calculation of appropriate expressions called <<transition metrics>>, or <<branch metrics>>, indicated in the figure with λ_(n) ⁽¹⁾, λ_(n) ⁽²⁾, λ_(n) ⁽³⁾, . . . , λ_(n) ^((SM′)). Said branch metrics reach the inputs of a block called Viterbi Processor, krown to the skilled in the art, giving at output the estimated sequence {â_(n)}. How this takes place in presence of a signal r(t) given by the (1) where c_(i), symbols appear, shall be explained describing some actual coding cases. Notice in the figure, that complex quantities are indicated with line arrows thicker than those used for real quantities.

[0054] The FRIC filter is matched to the real pulse transmitted h(t), of known trend, is a filter with response to pulse h(−t). In general an impulse h(t) is used for which the pulse coming out from the adapted filter meets the Nyquist condition of intersymbol interference absence, that is such that: $\begin{matrix} {{\int_{- \infty}^{\infty}{{h\left( {t - {iT}} \right)}\quad h\quad \left( {t - {kT}} \right){t}}} = \left\{ \begin{matrix} 1 & {{{for}\quad k} = i} \\ 0 & {otherwise} \end{matrix} \right.} & (2) \end{matrix}$

[0055] The selection of a FRIC filter with raised cosine root frequency response, likewise the transmission filter FTRAS, involves a best filtering during receipt and the matching of the (2). Therefore samples x,, can be expressed as: $\begin{matrix} \left( {{x_{n}\overset{\Delta}{=}\quad {{r(t)} \otimes {h\left( {- t} \right)}}}} \right)_{t = {nT}} & (3) \end{matrix}$

[0056] where the symbol

represents the convolution operator, and recalling the (1) and (2):

x _(n) =c _(n) e ^(jθ)+η_(n)  (3′)

[0057] having indicated with: $\begin{matrix} {\eta_{n} = {\int_{- \infty}^{\infty}{{w(t)}{h\left( {t - {nT}} \right)}{t}}}} & (4) \end{matrix}$

[0058] the noise samples filtered by the FRIC filter. As in the case of known phase, the samples x_(n) represent a sufficient statistic (that is their sequence includes the whole information associated to the corresponding continuous signal), and shall be indicated hereafter by the term <<observable>>.

[0059] Concerning the operation of the RIC receiver of FIG. 2, it is only necessary to describe the nature of the METRICTOT block, since the implementation of the Viterbi Processor block is known to the skilled in the art, saying first that the algorithm there developed searches the path according to maximum accumulated metric on a trellis sequential diagram, called also trellis, having S states, where M′ branches start from each one of them, each one characterized by its own metric, representing the probability associated to the occurrence of the specific transition among subsequent states of the trellis.

[0060] The structure of the METRICTOT block is better detailed in FIG. 3, where SM′ identical blocks METRIC(s) at N inputs for samples x_(n−1), n_(n−2), . . . , x_(n−N+1) can be noticed and, at an output for a relevant branch metrics λ_(n) ⁽³⁾. The single METRIC(s) blocks work all in parallel and differ only for the content of storage internal elements containing the coded symbols {tilde over (c)}_(n),{tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1) univocally associated to the particular branch of the trellis of which the relevant METRIC(s) block calculated the metric λ_(n) ^((s)).

[0061] In the RIC non-coherent receiver of the example not representing a restriction for the first embodiment of the invention, the analytical expression for the calculation of the metrics λ_(n) ^((s)) is obtained starting from the analytical expression of the metrics calculated by the coherent receiver. This last, assuming the ideal channel and the <<(best>> global filtering, selects the code sequence that maximises the sum of metrics having the following expression:

Re{x _(n) {tilde over (c)} _(n) ^(*) e ^(−jθ)}−½|{tilde over (c)} _(n)|²  (5)

[0062] where Re{.} indicates the real portion of a complex number {.}; {tilde over (c)}_(n) are univocally coded symbols associated to the generic branch of the trellis at the discrete n-th instant, the asterisk * indicating the total conjugate value of {tilde over (c)}_(n); and θ is the known phase of the modulated carrier. The different passages leading to the (5) are shown in tnhe volume under the title <<Digital communications>>, author J. Proakis, published by McGraw-Hill, New York 1989.

[0063] In the (5), the phasor e^(−jθ) is of course unknown to the non coherent receiver RIC, however in the implementation of the RIC receiver according to the present invention the (5) is used, replacing said phasor by its estimate based on the observation at the instant n, of N−1 samples preceding xn, according to the relation: $\begin{matrix} {\overset{\bigwedge}{^{{- j}\quad \theta}} = \frac{\sum\limits_{i = 1}^{N - 1}{x_{n - 1}^{*}c_{n - 1}}}{{\sum\limits_{i = 1}^{N - 1}{x_{n - i}^{*}c_{n - i}}}}} & (6) \end{matrix}$

[0064] where the symbol

represents said estimate of the phasor, to obtain which, the

−1 samples x_(n), differing from those bearing the same name in the (5) due to the fact they have been obtained through non coherent conversion in base band of the signal received, are stored to the purpose in a phase reconstruction memory represented by the scroll register SHF1 of FIG. 2.

[0065] Said estimator could replace the phasor e^(−jθ) in the (5), whenever data {tilde over (c)}_(n),{tilde over (c)}_(n−1), . . . , {tilde over (c)}_(n−N+1) already <<decided>> by the Viterbi processor are known, while the (6) contains data c_(n), c_(n−1), . . . ,c_(n−N+1) transmitted and therefore not yet known. The estimate of these last is fumished by the Viterbi Processor with an inevitable delay, which would consequently involve at the discrete n-th instant an incorrect value of the phasor e−jθ, and therefore unusable in the (5). To solve the problem to avail of the correct data sequence, the <<Per-Survivor Processing”, or PSP, technique can be used, described in the article <<Per-survivor processing: a general approach to MLSE in uncertain environments”, authors: R. Raheli, A. Polydoros and C. K. Tzou, published on IEEE Trans. Commun., vol. 43, pp.354-364, February/March/April 1995. Through this technique the Viterbi algorithm can be used in presence of an unknown parameter, in this case the phase θ, which if known, could be advantageously used in the expression of the branch metric. The measure consists in using the above mentioned parameter in the expression of the branch metric, estimated according to a sequence of samples x_(n −1), . . . ,x_(n−N+1) of the signal r(t), filtered by FRIC, and of a generic sequence of data {tilde over (c)}_(n),{tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1).

[0066] Operating as above the following analytical expression is obtained for the calculation of branch metrics in of the RIC receiver: $\begin{matrix} {\lambda_{n} = {\frac{{Re}\left\{ {\sum\limits_{i = 1}^{N - 1}{x_{n}x_{n - 1}^{*}{\overset{\sim}{c}}_{n}^{*}{\overset{\sim}{c}}_{n - 1}}} \right\}}{{\sum\limits_{i = 1}^{N - 1}{x_{n - 1}^{*}{\overset{\sim}{c}}_{n - 1}}}} - \frac{{{\overset{\sim}{c}}_{n}}^{2}}{2}}} & (7) \end{matrix}$

[0067] where {tilde over (c)}_(n),{tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1) are the univocally coded symbols associated to the generic branch of the trellis at the discrete n-th instant, and x_(n−1),x_(n−2), . . . ,x_(n−N+1) are N sequential samples of the signal received, non-coherently converted in base band and filtered at best. As it can be noticed from the (7), due to estimate (6), the phase θ does no more explicitly appear in the expression of the metric. In a manner completely equivalent to the concepts described above, we can say that the estimate of the phase reference implemented in PSP mode is <<aided by data>>, since for each path survived to the instant n, the N−1 data relevant to its previous history are used. In the hypothesis not to discretionary reduce the number of states of the trellis, when transition metrics λ_(n) ^((s)) are calculated at the instant n and the same are accumulated to select the survivors, the particular sequences {tilde over (c)}_(n){tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1) stored in the relevant METRIC(s) blocks of FIG. 3 used for the calculation of said metrics, correspond also to the N−1 data relevant to the previous history of those survivors. In consequence of the above, the estimate (6) of phasor e^(−jθ) in PSP modality does not involve any variation in the traditional application of the Viterbi algorithm, thus justifying what said above on full compatibility.

[0068] Of course the length N of the phase reconstruction memory is the result of a compromise between the need to make a more accurate estimate of the phasor present in the (5), the accuracy increasing as N increases, and that, not to excessively expand the size of the trellis of the non coherent receiver scope of the invention, in respect with the trellis of the coherent receiver used as starting point for the calculation of the metrics, this last being of course without phase reconstruction memory.

[0069] The (7) is the expression of a branch metric of the non-coherent receiver, which makes the calculation of path metric recurrent, perfectly compatible with the theoretical assumptions of the Viterbi algorithm, contrarily to what took place for the conventional non-coherent receivers. This new and original result, was possible because we started from the expression (5) of the branch metric of the best coherent receiver, which is compatible with Viterbi, but inevitably inherits the ideal character distinguishing a similar receiver, and is not suitable at all, to the use in a non coherent receiver like that of the invention. The modifications of this expression (5) made according to the instructions of the PSP technique have removed the ideal character from the metric expression (7) characterizing the present invention, while maintaining the actual operational compatibility features with the Viterbi algorithm and with the theoretical justifications laying at ground. It must be pointed out that the (7) is not the result of a mere application of the known PSP technique, but a combination of different operational phases founded on the fact to use the best expression of the metric of the coherent receiver to obtain a similar expression valid for the non coherent receiver. In the opinion of the applicant, nothing similar was seen until now, nor the teaching of the present invention could be obvious to the skilled in the art, whose knowledge is well represented by the various papers mentioned herein. The assumption at the base of the known expression of the branch metric (5), or of communication channel without intersymbol interference, is inevitably transferred to the expression of the branch metric (7), for the way it has been obtained. In the practice, this situation occurs with a good approximation in the channels made of low or medium capacity radio links, or in satellite connections. As it can be also noticed from the (7), the expression of the branch metric is absolutely general as for the type of coding, in fact, no rule has been specified up to now; in the practice a commonly adopted coding is the differential one.

[0070] Making reference to FIG. 4 it is now better detailed the structure of the generic block METRIC(s) of FIG. 3, which calculates the branch metric λ_(n) ^((s)) calculated using the expression (7). For a better reading of the diagram shown, arrows of different thickness are indicated in the figure, associated to real or complex quantities. The figure also shows at bottom the PROD and DIV generic blocks with the indication of the relevant operations accomplished. In particular the PROD block makes the product of the values at its inputs after having conjugated the value at input marked with *, and the DIV block makes the quotient between the values at its inputs. As it can be noticed, the generic samples x_(n) x_(n−1), . . . , x_(n−N+1) reach a first input of relevant multiplication blocks PROD, at the second input of which arrive the corresponding coded symbols {tilde over (c)}_(n),{tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N−1) present in a memory of the METRIC(s) block. Except for a first product x_(n) {tilde over (c)}_(n), all the remaining products x_(n−1){tilde over (c)}_(n−1) ^(*) are sent to a same number of inputs of an adder 3, whose outgoing sum is sent to a first input of an additional PROD block, at the second input of which arrives the first product x_(n){tilde over (c)}_(n). The product coming out from this last multiplier is sent to a REAL block that extracts the real part and sends it to a first input of a divider block DIV, at the second input of which arrives a value coming out from a MOD block. The input of the MOD block is reached by the sum coming out from the adder, of which the MOD block extracts the module and makes it available at output. The quotient coming out from the DIV block is sent to a first input of an adder 4, at the second input of which the value $- \frac{{{\overset{\sim}{c}}_{n}}^{2}}{2}$

[0071] arrives, obtained from the coded symbol {tilde over (c)}_(n) stored. The real value coming out from the adder 4 is the branch metric λ_(n) ^((s)) calculated by the METRIC(s) block.

[0072] It is now described the development of the (7) through the configuration of the blocks of to FIG. 4. First of all, the PROD blocks at top of the figure calculate the products x_(n){tilde over (c)}_(n) ^(*) and x_(n−i){tilde over (c)}_(n−i) ^(*), afterwards the adder 3 calculates the term ${\sum\limits_{i = 1}^{N - 1}{x_{n - 1}^{*}{\overset{\sim}{c}}_{n - 1}}},$

[0073] of which the MOD block subsequently calculates the module. The module just calculated could also be used in the (7) as divider, since the following relation is valid ${{\sum\limits_{i = 1}^{N - 1}{x_{n - 1}^{*}{\overset{\sim}{c}}_{n - 1}}}} = {{\sum\limits_{i = 1}^{N - 1}{x_{n - 1}^{*}{\overset{\sim}{c}}_{n - 1}}}}$

[0074] The term $\sum\limits_{i = 1}^{N - 1}{x_{n - 1}{\overset{\sim}{c}}_{n - i}^{*}}$

[0075] is also conjugated and then multiplied by x_(n){tilde over (c)}_(n) ^(*). From the product result, the real part is extracted, which divided by $\left. {\sum\limits_{i = 1}^{N - 1}{x_{n - 1}^{*}{\overset{\sim}{c}}_{n - i}}} \right|$

[0076] and summed to ${- \frac{{{\overset{\sim}{c}}_{n}}^{2}}{2}},$

[0077] forms the branch metric λ_(n) ^((s))

[0078] Concerning the particular coding of information symbols {a_(n)} in coded symbols {c_(n)} the maximum general character has been maintained up to now. We shall now illustrate some applications of the (7) in different coding conditions pertaining to the practical cases of larger use, highlighting additional inconveniences of the known art. The cases examined in detail are the following:

[0079] A) Non coherent sequence estimation receiver for signals with M-PSK modulation, transmitted together with pilot symbols as an alternative to differential coding.

[0080] B) Non coherent sequence estimation receiver for signals with M-PSK modulation and differential coding (M-DPSK); the same receiver where the maximum likelihood sequence estimation receiver takes place according to a simplified method is considered as sub-case.

[0081] C) Non ceherent sequence estimation receiver for signals with M-PSK modulation and channel convolutional coding.

[0082] D) Non coherent sequence estimation receiver for signals with M-QAM modulation and quadrant differential coding (M-DQAM).

[0083] Additional cases not specified, for instance the use of a TCM coding (Trellis Coded Modulation), can be easily drawn from examples A), B), C), and D) that shall be now described.

[0084] In the introduction, we said that non coherent receivers presuppose the use of a differential coding during transmission, due to the practical difficulty in the recovery of an absolute phase recovery. However, this statement is not binding since, as it is known, as an alternative to the differential coding, the above mentioned recovery can be made through periodical introduction of one or more pilot symbols known to the receiver in the information symbol sequence transmitted, every P of said symbols.

[0085] In the cases A), B) and C) the symbols {C_(n)} belong to the M-PSK alphabet, therefore the generic symbol transmitted can be expressed asc_(n)=e^(jφ) _(n) , where, ${\varphi_{n} \in \left\{ {{\frac{2\quad \pi \quad m}{M};{m = 0}},1,\quad \ldots \quad,{M - 1}} \right\}},$

[0086] with M cardinality of the alphabet. In the above mentioned cases the branch metric (7) can be simplified, in fact, being |c_(n)|=1, it is possible to define said branch metrics as: $\begin{matrix} {\lambda_{n} = \frac{{Re}\left\{ {\sum\limits_{i = 1}^{N - 1}{x_{n}x_{n - i}^{*}{\overset{\sim}{c}}_{n}^{*}{\overset{\sim}{c}}_{n - i}}} \right\}}{{\sum\limits_{i = 1}^{N - 1}{x_{n - i}^{*}{\overset{\sim}{c}}_{n - i}}}}} & (8) \end{matrix}$

[0087] since the term −½ appears in the same way in all the branches of the trellis. It has also been verified that neglecting the denominator, an improvement of the RIC receiver performance is obtained, in this case the (8) becomes: $\begin{matrix} {\lambda_{n} = {{Re}\left\{ {\sum\limits_{i = 1}^{N - 1}{x_{n}x_{n - i}^{*}{\overset{\sim}{c}}_{n}^{*}{\overset{\sim}{c}}_{n - i}}} \right\}}} & (9) \end{matrix}$

[0088] Using the (9), the structure of the METRIC(s) block of FIG. 4 of course can be simplified, in fact it is possible to directly obtain the branch metric λ_(n) ^((s)) at the output of the REAL block, thus eliminating the surplus blocks MOD, DIV, and the adder 4.

[0089] In the receiver of case A) symbols c_(n) shall be intended as information symbols a_(n) belonging to the M-PSK alphabet. The state of the trellis is defined as:

σ _(n)=({tilde over (c)}_(n−1),{tilde over (c)}_(n−2), . . . ,{tilde over (c)}_(n−N+1))  (9′)

[0090] and the number of states is S=M^(N−1), and therefore it exponentially increases with the phase reconstruction memory. If at the instant n a pilot symbol is received, for instance χ, the states compatible with the same are all of the type:

σ _(n)=(χ,{tilde over (c)}_(n−2), . . . ,{tilde over (c)}_(n−N+1))  (9″)

[0091] that is only one fraction 1/M of total states. In this case the Viterbi processor maintains only a fraction of the paths survived in the trellis, namely, only the survivors leading to the states (9″) compatible with the known pilot symbol. From the realization point of view, when the pilot symbol is received, the cumulative metrics of survived paths that shall be eliminated, can be decreased of an appropriate quantity causing that all the considered paths, generated by said survivors, loose all the subsequent comparisons. In conclusion, the receiver of case A) exploits its knowledge in advance, that is the fact to know that at a given instant k the symbol c_(n)=χ could not be transmitted, to avoid to have survivors ending in states not compatible with the known pilot symbol, and therefore reduce the possibility to decide for a wrong sequence.

[0092] The operation performance of a receiver implemented according to the case A) of the present invention, with use of pilot symbols as an alternative to differential coding, have been compared with those of an ideal coherent receiver which of course does not require pilot symbols. From the comparison, it resulted that for a phase reconstruction memory with N=4 and appropriate values of the cadence P of pilot symbols, the BER of the non-coherent receiver differs of max. 0,3 dB from that of the coherent receiver, for all the values considered of the signal-to-noise ratio.

[0093] It is now examined the case B) relevant to a non-coherent sequence estimation receiver for signals with M-PSK modulation and differential coding (M-DPSK). Symbols c_(n) are supposed to come, through a differential coding c_(n)=c_(n−1)a_(n), from a source of symbols a_(n) equipossible and independent, belonging to the M-PSK alphabet. Symbols {tilde over (c)}_(n) in the expression of branch metrics (9) can be expressed according to source symbols ã_(n) as follows: $\begin{matrix} {{{\overset{\sim}{c}}_{n}^{*}{\overset{\sim}{c}}_{n - i}} = {\prod\limits_{m = 0}^{i - 1}{\overset{\sim}{a}}_{n - m}^{*}}} & (10) \end{matrix}$

[0094] which introduced in the (9) gives for λ_(n): $\begin{matrix} {\lambda_{n} = {{Re}\quad \left\{ {\sum\limits_{i = 1}^{N - 1}{x_{n}x_{n - i}^{*}{\prod\limits_{m = 0}^{i - 1}{\overset{\sim}{a}}_{n - m}^{*}}}} \right\}}} & (11) \end{matrix}$

[0095] From the (11) we can notice how in the metric λ_(n) the coded symbols {tilde over (c)}_(n) do no more appear, therefore it is the Viterbi processor itself that without any modification in its operation mode, accomplishes also the differential, according to what shown in FIG. 2. To obtain this, and as it results from the (11), we see that the state σ_(n) of the trellis used by the Viterbi processor must be defined on information symbols as: σ_(n)=(ã_(n−1),ã_(n−2), . . . ,ã_(n−N+2))  (12)

[0096] The number of states is S=M^(N−2), whose exponential growth with N can be reasonably confined using small N values, because we observed that some performance can however be obtained, expressed through the BER, very close to those of a coherent receiver for signals with differential coding.

[0097] As sub-case B) it is described a non-coherent receiver with expression of the branch metric (11), differing from the previous one due to the fact that the maximum likelihood sequence estimation takes place according to a simplified method. Techniques are known in the invention application field enabling to reduce the number of states of the trellis on which the Viterbi processor operates. Generally, once the length of the phase reconstruction memory is determined, a given number of information symbols will appear in the definition of the deriving state; now, using the above mentioned known complexity reduction techniques it is possible to define a reduced trellis where the state is tied to a lower number of information symbols, for instance neglecting the more remote symbols, or making a partitioning of the symbol set (set partitioning). The above mentioned techniques are, for example, described in the following articles:

[0098] <<Reduced-State Sequence Estimation (RSSE), with set partitioning and decision feedback>>, authors: M. V. Eyuboglu, S. U. H. Qureshi, published on IEEE Trans. Commun., vol. 36, pp.13-20, January 1988;

[0099] <<Decoding of trellis-encoded signals in the presence of intersymbol interference and noise>>, authors: P. R. Chevillat and E. Eleftheriou, published on IEEE Trans. Commun., vol. 43, pp.354-364, July 1989.

[0100] The reduced complexity technique applied in the receiver of the present sub-case B) is of the RSSE type “reduced state sequence estimation”. The “reduced” state is defined as on σ_(n)=(ã_(n−1),ã_(n−2), . . . ,ã_(n−Q+2)), being the integer Q≦N. In this way the number of states of the trellis diagram becomes S=M^(Q−2). The remaining information symbols α_(n−Q+1), . . . ,α_(n−N+2) necessary to the estimate of the phasor e^(−jθ) according to (6), not included, or not completely specified in the definition of reduced state, are found in PSP mode from the previous history of each survivor. A similar approach is described in section IV.A of the mentioned article by R. Raheli, A. Polydoros and C. K. Tzou, of February/March/April 1995.

[0101] This technique (PSP) of reduction of the number of states, can be applied also to the previous case A) and to the subsequent cases C) and D).

[0102] The performance of the non-coherent receiver of case B), and relevant sub-case, have been evaluated for a 4-DPSK modulation (DQPSK, from Differential Quaternary PSK). Variable lengths of the phase reconstruction memory have been considered and therefore, different complexity levels of the trellis. The receiver according to the present invention has higher performance, at equal N value, compared to the N-differential receiver described in the article by Divsalar and M. K. Simon of March 1990. For N=2 both the receivers degenerate in the classical differential receiver, so their performances coincide.

[0103] It was also checked that the performance of the receiver considered tend, as N increases, to those of a coherent receiver of the DQPSK type, with a rapidity not depending on the signal-to-noise ratio. This demonstrates that not only asymptotically, for high signal-to-noise ratios, but in general for each value of this ratio, the performance of a coherent receiver can be approximated as desired, provided that a sufficiently high N value is chosen. This is true also when the complexity is appropriately reduced. For instance, it results that with a not excessive complexity (reduced to 16 states) the performance decrease is rather limited versus the coherent receiver (0,2 dB at a BER of 10⁻⁴). Therefore the receiver scope of the present invention, in the application referred to case B) is characterized by a insignificant, loss compared to a coherent receiver, loss that in some applications can be lower than that due to an inaccurate estimation of the phase in the approximation of a coherent system through a pseudocoherent receiver.

[0104] It is now examined the case C) that concerns a non-coherent sequence estimation receiver for signals with M-PSK modulation and channel convolutional coding. The following considerations are applicable to any type of channel coding, for instance to the TCM coding. To the purpose of better facilitate the comprehension of the structure and the operation of the receiver relevant to the case C), making reference to FIG. 5, it is first shown a convolutional coder used in the TRAS transmitter of FIG. 1 for the case considered. The structure of the coder is that described in the paper by D. Raphaeli of February 1996 mentioned above. At the input of the coder shown in the figure, one can notice a sequence of information symbols {δ_(n)}, with 1/T cadence, belonging to an alphabet A={0,1, . . . , M−1 }. The sequence {δ_(n)} is sent to a chain of K delay elements τ₁′,τ₂′, . . . ,τ_(N)′ of a symbol interval T. These elements τ′ are the flip-flops of a scroll register SHF2 storing a string of K said symbols for the whole duration of the symbol, making them available at the same time at the output of each single flip-flop τ′. Each sample δ_(k) coming out from a relevant flip-flop τ′ simultaneously reaches a first input of η multipliers

_(ij); at the second input of said multipliers arrive the relevant constants g_(ij)∈ A. From the above, the total number of multipliers

_(ij) will result K_(η), organized in η groups of K multipliers each. Indexes i and j placed as deponent of elements

_(ij) and g_(ij) indicate respectively the element i within the group of K elements, and a particular group j within the assembly of η groups. The outputs of the K multipliers

_(ij), within each group j, reach the K inputs of a relevant adder Σ_(j), module M. Each output of the η adders Σ_(j) reaches a relevant input of a selector SEL which, with cadence equal to η times the symbol frequency 1/T, cyclically withdraws a sample at the selected input and sends it to a mapping block MAP. The samples coming out from the selector SEL form a sequence {ε_(ηk+1)} ∈ A (I=0,1, . . . , η−1), which is mapped at the output by MAP in a corresponding complex sequence {c_(ηk+1)} of the M-PSK alphabet, as indicated in figure by the different thickness of the arrows. The notation used herein foresees that the index k represents a time instant (discrete) synchronous with the information sequence and the index I defines each one of the η coded symbols associated to each information symbol.

[0105] As for the operation of the coder in FIG. 5, the value K represents the length of code limit, which is characterized by a number of states equal to S_(c)=M^(K−1). The constants g_(ij) ∈ A, are organized in η K-uple g₁=(g₁₁, . . . ,g_(K1)) . . . g_(η)=(g_(1η), . . . , g_(Kη)), and form the code generators, having rate 1/η. It is then possible to express the symbols ε_(ηk+1) coming out from the selector SEL as follows: $\begin{matrix} {ɛ_{{\eta \quad k} + 1} = {\sum\limits_{m = 1}^{K}{g_{ml}\delta_{k - m}}}} & (13) \end{matrix}$

[0106] where the summation has to be intended “module M”. The mapping operation is made by the MAP block through the following relation: $c_{{\eta \quad k} + l} = {^{j\frac{2\pi}{M}s_{{\eta \quad k} + l}}.}$

[0107] It is possible to define a relation between symbols c_(ηk+1) and the a_(n) remembering that lacking the convolutional coding, the symbols δ_(n) should be mapped by the MAP block in symbols a_(n) belonging to the M-PSK alphabet according to the relation: $a_{n} = {^{j\frac{2\pi}{M}\delta_{k}}.}$

[0108] With adequate treatment of the previous expressions, we obtain: $\begin{matrix} {c_{{\eta \quad n} + l} = {\prod\limits_{m = 1}^{K}a_{n - m}^{g_{ml}}}} & (14) \end{matrix}$

[0109] This assumed, the calculation of branch metrics concerning the receiver of the present case C) that employs the coder of FIG. 5, can be made starting from the expression (9) of the branch metric for the similar receiver M-PSK, without channel coding. In the case of the present convolutional coding, the index n in (9) can be replaced by ηn+j, where the new index n scans the information symbols, while the index j scans the η coded symbols associated to the n-th information symbol. Also, the Viterbi algorithm operates on information symbols and therefore the contribution to the branch metrics relevant to the η coded symbols corresponding to a same information symbol must be summed up. Consequently, the branch metrics (9) become: $\begin{matrix} {\lambda_{n} = {{Re}\quad \left\{ {\sum\limits_{i = 1}^{L - 1}{\sum\limits_{j = 0}^{\eta - 1}{\sum\limits_{l = 0}^{\eta - 1}{x_{{\eta \quad n} + j}x_{{\eta \quad {({n - i})}} + l}^{*}{\overset{\sim}{c}}_{{\eta \quad n} + j}^{*}{\overset{\sim}{c}}_{{\eta \quad {({n - i})}} + l}}}}} \right\}}} & (15) \end{matrix}$

[0110] where L=N/η is assumed as integer. Using the (14) in (15) it is possible to express code symbols according to information symbols, obtaining: $\begin{matrix} {\lambda_{n} = {{Re}\quad \left\{ {\sum\limits_{i = 1}^{L - 1}{\sum\limits_{j = 0}^{\eta - 1}{\sum\limits_{l = 0}^{\eta - 1}{x_{{\eta \quad n} + j}{x_{{\eta \quad {({n - i})}} + l}^{*}\left( {\prod\limits_{m = 1}^{K}{\overset{\sim}{a}}_{n - m}^{g_{mj}}} \right)}^{*}{\prod\limits_{k = 1}^{K}{\overset{\sim}{a}}_{{({n - i})} - k}^{g_{kl}}}}}}} \right\}}} & (16) \end{matrix}$

[0111] which is the expression of a branch metric for the receiver considered, whose trellis has a riumber of states S=S_(c)M^(L−1), where S_(c)=M^(K−1), and therefore S=M^(L+K−2), having implicitly considered the case not requiring the introduction of a differential coding before the channel coder: this position shall be clarified hereafter. In the contrary instance the number of states is S=S_(c)M^(L−2) due to reasons similar to those expressed in the calculation of the number of states of the receiver M-DPSK of case B). Since the trellis results rather complex, it can be the case to apply the described complexity reduction techniques.

[0112] Making reference to FIGS. 6 and 7, we shall now describe some known subjects, concerning in general the channel codes expressed by the coder COD of FIG. 1; the purpose is that to determine the advantages for the receivers according to the invention, by comparison with conventional ones.

[0113] Two blocks 5 and 6 placed in cascade are shown in FIG. 6, which represent a particular implementation of the coder COD included in the transmitter TRAS of FIG. 1. The block 5 represents a differential coder, the input of which is reached by the sequence of information symbols a={a_(n)}, and a sequence of symbols b={b_(n)} coded in differential mode is supplied at its output. The block 6 is a channel coder expressing a phase rotation invariant code of multiples of an angle φ sustained by vector b representing the sequence carrying the same name. At the output of block 6 the sequence c={c_(n)}is present. Both the coders are of the known type. For the coder 6 the following property applies, that is described in the volume under the title “Introduction to trellis-coded modulation with applications”, authors: E. Biglieri, D. Divsalar, P. J. McLane and M. K. Simon, published by Macmillan Publishing Company, 1991: if the coded sequence c={C_(n)} corresponds to the sequence b={b_(k)} of symbols at the input of the channel coder, the sequence ce^(jnφ), with any m, is still a code sequence and corresponds to an input sequence be^(jmφ). This code behaviour, lacking appropriate countermeasures, could result catastrophic for the receiver, since it would prevent the decoding of starting symbols. The countermeasures which make possible to use the same a code of this type, consist in placing a differential coding on information symbols before the channel coding, as shown in FIG. 6. In this case the sequences b and be^(jmφ) that are not discriminated by the decoder, are corresponding to a same information sequence a={a_(n)} through a differential coding, and the problem is thus solved. This enables to see how the cascade of the two blocks 5 and 6 of FIG. 6 behaves as a unique coder relevant to a code globally non-invariant to rotations, corresponding to the unique block 7 of FIG. 7.

[0114] The implementation of the COD block of FIG. 1 according to the setting of FIG. 6 makes possible in the known art the use of a pseudocoherent receiver, which employs the metrics of the coherent receiver assuming as true the phase obtained making use of a PLL, because the reception is possible even if the PLL locks not to the true phase but to a phase differing from the same of a multiple of the angle φ. In this case it is necessary to construct the trellis on the symbols {b_(n)}, except for the subsequent use of a differential decoder to obtain the decided symbols {a_(n)}. This double decoding passage can be avoided in the receivers of the subject invention. In fact, in presence of a rotation invariant code preceded by differential coding, the trellis is constructed not on the symbols {b_(n)} but on information symbols {a_(n)}; in this way the Viterbi processor makes also the differential decoding. If one wants to proceed in a different way and construct the trellis on coded symbols {tilde over (c)}_(n), the same path metric would be associated to the input sequences which differ for a constant phase rotation, due to the fact that products of the type {tilde over (c)}_(n) ^(*){tilde over (c)}_(n−i) invariant to phase rotation appear in the branch metrics (7). Therefore, more maximum likelihood paths would be present on the trellis. Then, such a choice must be rejected.

[0115] An additional advantage that non coherent receivers of the invention have, compared to the known art consisting of pseudocoherent receivers with PLL, is that to be able to freely use a channel code non invariant to phase rotations. In fact, as for the particular known art considered, we have just seen that the coder structure shown in FIG. 6 results binding, where the channel code is rotation invariant. On the other hand, a channel code non invariant to phase rotation (differently obtained from the cascade of blocks of FIG. 6) would not enable the pseudocoherent receiver to neutralize the phase ambiguity introduced by the PLL. To overcome this serious disadvantage, the solutions known consist in giving a phase absolute reference to the receiver, for instance, using the pilot symbols mentioned above during transmission.

[0116] Concerning the non coherent receiver of the present invention, a constant phase rotation of the carrier of any angle θ ∈ [0,2π] has no effect, considering that in the metrics of the receiver expressed in (7) appear products of the type x_(n)x_(n−i) ^(*) which delete the rotation θ. Therefore, also a code non invariant to rotation can be used according to the diagram of FIG. 7, without the need of using pilot symbols. However, this advantage is not exclusive of. the receiver according to the present invention, but belongs to all non-coherent receivers.

[0117] The performance of the non-coherent receiver of the present case C has been compared to that of the receiver described in the article by D. Raphaeli of February 1996 mentioned above. The parameters selected to characterise both the receivers, foresaw the use of signals with QPSK modulation and convolutional coding, phase rotation not invariant, with K=3 and η=2 (and therefore S_(c)=16). The code generators used to this purpose were: g₁=(1, 3, 3) and g₂=(2, 3, 1). A phase reconstruction memory has been used in the receiver with L=4. From the comparison it resulted that the receiver of the present case C) gains 0,3 dB in performance on BER, for any value considered of the signal-to-noise ratio, compared to the receiver of the known art, and loses only 0,2 dB compared to the best coherent receiver.

[0118] Finally, we describe the case D) relevant to a non-coherent sequence estimation receiver for signals with M-QAM modulation and differential coding (M-DQAM). For the following subjects, reference to FIGS. 1, 2, 3 and 4, still applies together with notions introduced describing the case B) relevant to a non-coherent sequence estimation receiver for signals with M-PSK modulation and differential coding (M-DPSK).

[0119] Concerning the receiver of the present case D), the symbols c_(n) belonging to the M-QAM alphabet are supposed to be derived from information symbols a_(n) of the same alphabet, equipossible and independent, through a quadrant differential coding. A set belonging to the M-QAM set can be expressed as follows:

a _(n)=μ_(n) p _(n)  (17)

[0120] where μ_(n) is the a_(n) symbol multiplied by a phasor multiple of π/2 which brings it in the first quadrant of the complex plan, and p_(n) belongs to the set QPSK {±1, ±j}; p_(n) represents just the reverse of the subject phasor. The quadrant differential coding produces a coded symbol:

c _(n)=μ_(n) q _(n)  (18)

[0121] with:

q =p _(n) q _(n−1)  (19)

[0122] which is in practice the usual differential coding for M-PSK modulations applied to symbols p_(n) of a constellation QPSK. Branch metrics defined by the more general expression (7) can be expressed depending on information symbols a_(n) in order that the Viterbi processor implements also the quadrant differential decoding. In particular, for (17), (18) and (19) we have: $\begin{matrix} {{{\overset{\sim}{c}}_{n}^{*}{\overset{\sim}{c}}_{n - i}} = {{{\overset{\sim}{\mu}}_{n}^{*}\mu_{n - i}{\overset{\sim}{q}}_{n}^{*}{\overset{\sim}{q}}_{n - i}} = {{{\overset{\sim}{\mu}}_{n}^{*}\mu_{n - i}{\prod\limits_{m = 0}^{i - 1}{\overset{\sim}{p}}_{n - m}^{*}}} = {{\overset{\sim}{a}}_{n}^{*}{\overset{\sim}{\mu}}_{n - 1}{\prod\limits_{m = 1}^{i - 1}{\overset{\sim}{p}}_{n - m}^{*}}}}}} & (20) \end{matrix}$

[0123] Multiplying the term ${{{\sum\limits_{i = 1}^{N - 1}{x_{n - i}^{*}{\overset{\sim}{c}}_{n - i}}}}\quad {by}\quad {{\overset{\sim}{q}}_{n}^{*}}} = 1$

[0124] and considering that: $\begin{matrix} {{{\overset{\sim}{q}}_{n}^{*}{\overset{\sim}{c}}_{n - i}} = {{{\overset{\sim}{\mu}}_{n - i}^{*}{\overset{\sim}{q}}_{n - i}{\overset{\sim}{q}}_{n}^{*}} = {{\overset{\sim}{\mu}}_{n - i}^{*}{\overset{\sim}{p}}_{n}^{*}{\prod\limits_{m = 1}^{i - 1}{\overset{\sim}{p}}_{n - m}^{*}}}}} & (21) \end{matrix}$

[0125] and also that |{tilde over (c)}_(n)|=|ã_(n)|=|{tilde over (μ)}_(n)| we obtain from the (7): $\begin{matrix} {\lambda_{n} = {\frac{{Re}\quad \left\{ {\sum\limits_{i = 1}^{N - 1}{x_{n}x_{n - i}^{*}{\overset{\sim}{a}}_{n}^{*}{\overset{\sim}{\mu}}_{n - i}{\prod\limits_{m = 1}^{i - 1}{\overset{\sim}{p}}_{n - m}^{*}}}} \right\}}{{\sum\limits_{i = 1}^{N - 1}{{\overset{\sim}{x}}_{n - i}^{*}{\overset{\sim}{\mu}}_{n - i}{\overset{\sim}{p}}_{n}{\prod\limits_{m = 1}^{i - 1}{\overset{\sim}{p}}_{n - m}^{*}}}}} - \frac{{{\overset{\sim}{a}}_{n}}^{2}}{2}}} & (22) \end{matrix}$

[0126] Therefore a trellis diagram can be defined with state:

σ_(n)=({tilde over (μ)}_(n−1),{tilde over (μ)}_(n−2), . . . ,{tilde over (μ)}_(n−N+1),{tilde over (p)}_(n−1),{tilde over (p)}_(n−2). . . , {tilde over (p)}_(n−N+2))=  (23)

=(ã_(n−1),ã_(n−2), . . . , ã_(n−N+2),{tilde over (μ)}_(n−N+1))

[0127] and the number of states results S=M^(N−1)/4.

[0128] The performance of the non coherent M-DQAM receiver of the present case D) have been compared with those of the receiver described in the paper by D. Divsalar and M. K. Simon of January 1994, mentioned above. From the comparison it resulted that the receiver scope of the invention is characterized by a BER lower than that of the to mentioned known art for any value considered of the signal-to-noise ratio, coming closer to the performance of the best coherent receiver with quadrant differential coding.

[0129] It is now considered the second embodiment of the present invention, starting from a first case where the signal received is characterized by linear modulations and absence of intersymbol interference. In this first case the filter FRIC (FIG. 2) is then suitable to the real pulse transmitted h(t), assumed as known, as it was up to now. For the following description it is convenient to better detail the (0.2) as follows: $\begin{matrix} {{s\left( {t,a} \right)} = {\sum\limits_{n = 0}^{N_{T} - 1}{c_{n}h\quad \left( {t - {nT}} \right)}}} & (0.3) \end{matrix}$

[0130] where N_(T) indicates the total number of code symbols transmitted.

[0131] In the noncoherent receiver RIC of the example, not limited to the same, referred to the second embodiment of the invention, the analytical expression for the calculation of metrics λ_(n) is obtained in a new way starting from the following known analytical expression of the sequence transmitted ã estimated at the maximum likelihood by the non coherent receiver: $\begin{matrix} {\hat{a} = {\underset{\overset{\sim}{a}}{\arg \quad \max}\left\{ {{- \frac{1}{2N_{0}}}{\int_{T_{0}}{{{\left( {{s\left( {t,\overset{\sim}{\left. a \right)}} \right.}} \right)^{2}{t}} + {\log \quad {I_{0}\left( {\frac{1}{N_{0}}{{\int_{T_{0}}{{r(t)}{s^{*}\left( {t,\overset{\sim}{a}} \right)}{t}}}}} \right)}}}}}} \right\}}} & (24) \end{matrix}$

[0132] where I₀(x) I₀(x) is the modified Bessel function of the first type and of order zero, T₀ is the observation interval, N₀ is noise the one-side power spectral density, and ã is a generic sequence of information symbols.

[0133] Such an expression for the sequence ã estimation is described, for instance, in Appendix 4C of the volume “DIGITAL COMMUNICATIONS”, author J. Proakis, published by McGraw-Hill, New York, 2nd ed., 1989.

[0134] In the case considered of linear modulations it is necessary to replace the (0.3) in the (24) and assume an observation interval T₀ long enough, afterwards, with simple algebraic passages the (24) becomes: $\begin{matrix} {\hat{a} = {\underset{\overset{\sim}{a}}{\arg \quad \max}\left\{ {{{- \frac{1}{2N_{0}}}{\sum\limits_{n = 0}^{N_{T} - 1}{{\overset{\sim}{c}}_{n}}^{2}}} + {\log \quad {I_{0}\left( {\frac{1}{N_{0}}{{\sum\limits_{n = 0}^{N_{T} - 1}{x_{n}{\overset{\sim}{c}}_{n}^{*}}}}} \right)}}} \right\}}} & (25) \end{matrix}$

[0135] where {{tilde over (c)}_(n)} is the code sequence univocally associated to the hypothetical sequence of information symbols ã according to the specified coding rule, and x_(n) is the output sampled at instant t=nT of a matched filter, as defined in the (3).

[0136] The (25) can be approximated assuming logI₀(x){tilde over (=)}x; the quality of the approximation being as better, at equal N₀, as N_(T) is larger.

[0137] More in general, let's first consider the modulations where the pulses of the modulated signal have different power among them, like in the case M-QAM, and define, starting from the (25) a general sequence metric like: $\begin{matrix} {{\Lambda_{N_{T}}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}{{\frac{1}{2}{\sum\limits_{n = 0}^{N_{T} - 1}{{\overset{\sim}{c}}_{n}}^{2}}} + \left( {{\sum\limits_{n = 0}^{N_{T} - 1}{x_{n}{\overset{\sim}{c}}_{n}^{*}}}} \right)}} & (26) \end{matrix}$

[0138] which can be obtained through recurrent updating of a partial sequence metric defined coinciding with the n-th signalling interval like: $\begin{matrix} {{\Lambda_{n}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}{{{- \frac{1}{2}}{\sum\limits_{k = 0}^{n - 1}{{\overset{\sim}{c}}_{k}}^{2}}} + \left( {{\sum\limits_{k = 0}^{n - 1}{x_{k}{\overset{\sim}{c}}_{k}^{*}}}} \right)}} & (27) \end{matrix}$

[0139] the (27) being in its turn possible to be obtained as accumulation of incremental metrics Δ_(n)(ã): $\begin{matrix} {{\Lambda_{n}\left( \overset{\sim}{a} \right)} = {\sum\limits_{k = {- n}}^{n}{\Delta_{k}\left( \overset{\sim}{a} \right)}}} & (28) \end{matrix}$

[0140] the incremental metric Δ_(n)(ã):can therefore be calculated from the (27): $\begin{matrix} {{\Delta_{n}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}{{{\Lambda_{n + 1}\left( \overset{\sim}{a} \right)} - {\Lambda_{n}\left( \overset{\sim}{a} \right)}} = {{{\sum\limits_{k = 0}^{n}{x_{k}{\overset{\sim}{c}}_{k}^{*}}}} - {{\sum\limits_{k = 0}^{n - 1}{x_{k}{\overset{\sim}{c}}_{k}^{*}}}} - {\frac{1}{2}{{{\overset{\sim}{c}}_{n}}^{2}.}}}}} & (29) \end{matrix}$

[0141] The difficulty in the calculation of the incremental metric (29) is a consequence of the non-limited memory necessary to express the same. Said metric depends in fact on the whole previous coded sequence, and the maximization of the general sequence metric would necessarily involve a research on a duly defined tree diagram. This research is reasonably feasible only when the length of the sequence transmitted consists of a few symbols, in the contrary instance the exponential growth of the number of the tree branches at each new symbol transmitted would quickly render this research unlikely to be proposed. In the second embodiment of the present invention, the inconvenient just highlighted is avoided through a suitable limitation of the memory of the incremental metric which, at the expenses of a neglectable information loss enables to maximize the sequence general metric performing a research on a trellis diagram rather than on a tree diagram.

[0142] The deriving advantage is significant since in the case of research on a trellis the number of branches remains constant at each symbol interval n-th, contrarily to what happens on a tree, then the known Viterbi algorithm is well applicable.

[0143] To limit the memory of the incremental metric, a truncation is introduced in the (29) such that only the most recent N observable xe_(k) are considered, being N<<N_(r), and the corresponding code symbols{tilde over (c)}_(k). After the starting transient state, that is, for n≧N−1, the resulting branch metrics, obtained from the expression (29) through memory truncation, will be: $\begin{matrix} {{\lambda_{n}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}{{{\sum\limits_{k = 0}^{N - 1}{x_{n - i}{\overset{\sim}{c}}_{n - i}^{*}}}} - {{\sum\limits_{k = 0}^{N - 1}{x_{n - i}{\overset{\sim}{c}}_{n - i}^{*}}}} - {\frac{1}{2}{{{\overset{\sim}{c}}_{n}}^{2}.}}}} & (30) \end{matrix}$

[0144] As in the case of the first embodiment of the invention, it is necessary to provide in the receiver a memory of N positions for a same number of samples x_(n), the above mentioned memory corresponds to the scroll register SHF1 of FIG. 2, representing a conception continuity element between the two embodiments. As it can be also noticed, the expression of the branch metric in both the embodiments includes a sum of products among the observable N stored and of the corresponding code symbols associated to the transitions on the trellis.

[0145] Notice that the approximation logI₀(x){tilde over (=)}x could be avoided, in this case the following branch metrics would be obtained: $\begin{matrix} {{\lambda_{n}^{\prime}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}{{\log \quad {I_{0}\left( {\frac{1}{N_{0}}{{\sum\limits_{k = 0}^{N - 1}{x_{n - i}{\overset{\sim}{c}}_{n - i}^{*}}}}} \right)}} - {\log \quad {I_{o}\left( {\frac{1}{N_{0}}{{\sum\limits_{k = 0}^{N - 1}{x_{n - i}{\overset{\sim}{c}}_{n - i}^{*}}}}} \right)}} - {\frac{1}{2N_{0}}{{\overset{\sim}{c}}_{n}}^{2}}}} & (31) \end{matrix}$

[0146] depending of course on the signal-to-noise ratio. Even if the logI₀(x){tilde over (=)}x approximation is not present in these metrics, the receivers based on the same do not necessarily give a performance better than those of the receivers based on metrics (30). In fact the effect of the truncation on the most recent symbols can be different in the two cases. It has been verified that the receivers based on the metrics (31) have performance equivalent to that of the receivers based on metrics (30) in all the cases considered below.

[0147] An additional conception continuity element with the first embodiment is obtained considering the modulations where the pulses of the modulated signal have equal power, as it occurs in M-PSK modulations; in this case the following description will lead to obtain for the branch metric just the same expression (9) previously obtained from the first embodiment, concerning the similar case. Reconsidering to this purpose the (25) we see that the approximation logI₀(x){tilde over (=)}x is no more necessary since the first term can be omitted, being constant for all the sequences, and the function logI₀(x) is growing monotone for x≧0 In this case therefore, the sequence general metric becomes: $\begin{matrix} {\hat{a} = {\underset{\overset{\sim}{a}}{\arg \quad \max}{{\sum\limits_{n = 0}^{N_{T} - 1}{x_{n}{\overset{\sim}{c}}_{n}^{*}}}}}} & (32) \end{matrix}$

[0148] and the relevant expression of the branch metric can be obtained operating as done before to come to the (30); with the difference that now versus the (30) the last term will be absent and the sole approximation in the calculation shall be due to the truncation, resulting in excess the one given by logI₀(x){tilde over (=)}x. Considering the fact that the function y=x² is growing monotone for x≧0, a sequence general metric equivalent to the (32) results being: $\begin{matrix} \begin{matrix} {{{\sum\limits_{n = 0}^{N_{T} - 1}{x_{n}{\overset{\sim}{c}}_{n}^{*}}}}^{2} = {{\sum\limits_{n = 0}^{N_{T} - 1}{\sum\limits_{m = 0}^{N_{T} - 1}{x_{n}x_{m}^{*}{\overset{\sim}{c}}_{n}^{*}{\overset{\sim}{c}}_{m}^{\sim}}}} =}} \\ {= {{\sum\limits_{n = 0}^{N_{T} - 1}{{x_{n}}^{2}{{\overset{\sim}{c}}_{n}}^{2}}} + {2\quad {Re}{\left\{ {\sum\limits_{n = 0}^{N_{T} - 1}{x_{n}{\overset{\sim}{c}}_{n}^{*}{\sum\limits_{m = 0}^{n - 1}{x_{m}{\overset{\sim}{c}}_{m}^{*}}}}} \right\}.}}}} \end{matrix} & (33) \end{matrix}$

[0149] In this expression the sequence general metric is given by the sum of all the elements of a hermitiane matrix N_(τ)×N_(τ). The first summation appearing in the (33) is independent from the coded sequence, being |{tilde over (c)}_(n)|²=1. Therefore an equivalent simplified expression of the metric (33) is: $\begin{matrix} {{\Lambda_{N_{T}}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}{{Re}{\left\{ {\sum\limits_{n = 0}^{N_{T} - 1}{x_{n}{\overset{\sim}{c}}_{n}^{*}{\sum\limits_{m = 0}^{n - 1}{x_{m}{\overset{\sim}{c}}_{m}^{*}}}}} \right\}.}}} & (34) \end{matrix}$

[0150] Operating on the (34) through a procedure similar to the one that conducted to the (30), we can define, by n≧N−1, the expression of a truncation incremental metric, which can be used to calculate the following branch metrics: $\begin{matrix} {{\lambda_{n}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}{{{Re}\quad \left\{ {x_{n}{\overset{\sim}{c}}_{n}^{*}{\sum\limits_{m = {n - N + 1}}^{n - 1}{x_{m}^{*}{\overset{\sim}{c}}_{m}}}} \right\}} = {{Re}\left\{ {\sum\limits_{i = 1}^{N - 1}{x_{n}x_{n - i}^{*}{\overset{\sim}{c}}_{n}^{*}{\overset{\sim}{c}}_{n - i}^{*}}} \right\}}}} & (35) \end{matrix}$

[0151] which is equivalent to the expression (9) above. The (30) and (35) enable the maximization of the sequence general metric recurrently operating through the Viterbi algorithm on a trellis whose branch metrics are in fact the (30) or its simplification (35) equivalent to the (9). As already highlighted, for a given coding rule, the coded symbols {c_(n)} can be expressed according to the information symbols {a_(n)} and the trellis state can be defined according to the same, thus avoiding the additional decoding. The (30) and (35) depend on N code symbols, in general, the number of states of the trellis of is the symbols is higher than the number of states of the code trellis. However, this complexity increase can be limited taking appropriate complexity reduction techniques, as the already mentioned ones, in order to limit the number of states without excessively reduce the value of N. In practice, it was possible to obtain a performance very close to that possible to obtain in case of ideal coherent detection (that is in the case the phase θ is perfectly known) of the phase, though using N values of some unit. This is a concrete demonstration of the scarce significance of the loss of information due to the cutting at N symbols the memory length for the calculation of the branch metrics. A possible explanation of the results obtained is that the main information to the purposes of the sequence estimation results concentrated in the most recent symbols.

[0152] Concerning the receiver used in the second embodiment of the present invention, within the limits of what described up to now, the schematization of the receiver RIC of FIG. 2 and the METRICTOT block of FIG. 3 still applies with the warning that the internal structure of the generic block forming the METRIC(s) of FIG. 3, will change to match the expression of the branch metric (30), while it will remain identical in the case the (35) is used. From the previous description of the internal structure of the METRIC(S) block of FIG. 4, and from the nature of the new expression (35), the skilled in the art could obtain the teaching to implement a METRIC(s) block suitable to the calculation of such an expression.

[0153] Remaining in the frame of signals affected by linear modulations and free from intersymbol interference, the cases A), B), C) and D) previously indicated are now examined in order and summarized below:

[0154] A) Sequence estimation non-coherent receiver for signals with M-PSK modulation, transmitted together with pilot symbols as an alternative to differential coding.

[0155] B) Sequence estimation non-coherent receiver for signals with M-PSK modulation and differential coding (M-DPSK).

[0156] C) Sequence estimation non-coherent receiver for signals with M-PSK modulation and channel convolutional coding.

[0157] D) Sequence estimation non-coherent receiver for signals with M-QAM modulation and quadrant differential coding (M-DQAM).

[0158] Concerning the case A) the considerations and the results of the first embodiment apply without any modification.

[0159] In case B) symbols {C_(n)} are derived from information symbols {α_(n)}, belonging to the same alphabet, through the differential coding rule c_(n)=c_(n−1)α_(n). Applying such a coding the branch metric (30) becomes: $\begin{matrix} {{\lambda_{n}\left( \overset{\sim}{a} \right)} = {{{\sum\limits_{i = 0}^{N - 1}{x_{n - i}{\prod\limits_{m = 0}^{i - 1}{\overset{\sim}{a}}_{n - m}^{*}}}}} - {{\sum\limits_{i = 1}^{N - 1}{x_{n - i}{\prod\limits_{m = 0}^{i - 1}{\overset{\sim}{a}}_{n - m}^{*}}}}}}} & (36) \end{matrix}$

[0160] having expressed the coded symbols {c_(n)} as function of symbols {α_(n)}. In the case of differential coding applied to a PSK alphabet at M points, it is necessary to operate on the branch metric (35), equivalent to the (9), which gives an expression identical to the (11 ) of the first embodiment. The state of the trellis valid for the (35) and (36) is given by the (12). The number of states is S=M^(N−2) exponentially depends on N, though also in this case the complexity reduction techniques mentioned above can be used.

[0161] As for the case C) reference shall be made to the considerations already made on the convolutional coding, in particular, the (14) continues to apply, reported here for convenience: $\begin{matrix} {c_{{\eta \quad n} + l} = {\prod\limits_{m = 1}^{K}a_{n - m}^{g_{ml}}}} & (14) \end{matrix}$

[0162] with the same variability of the indexes used. In this case the branch metrics (30) can be expressed in an equivalent manner in the form: $\begin{matrix} {{\lambda_{n}\left( \overset{\sim}{a} \right)} = {{{\sum\limits_{i = 0}^{{N/\eta} - 1}{\sum\limits_{l = 0}^{\eta - 1}{x_{{\eta {({n - i})}} + l}{\overset{\sim}{c}}_{{\eta {({n - i})}} + l}^{*}}}}} - {{\sum\limits_{i = 0}^{{N/\eta} - 1}{\sum\limits_{l = 0}^{\eta - 1}{x_{{\eta {({n - i})}} + l}{\overset{\sim}{c}}_{{\eta {({n - i})}} + l}^{*}}}}}}} & (37) \end{matrix}$

[0163] where the indexes n and i run on information symbols, l scans the code symbols associated to the (n−i)-th information symbol, and {fraction (N/η)} is comparable to the length L of the phase reconstruction memory expressed in term of the information symbols. Similarly, the branch metric (35), equivalent to the (9), is expressed as indicated in the (15) of the first embodiment.

[0164] Using the (14) in the (37) it is possible to express code symbols as function of information symbols, obtaining: $\begin{matrix} {{\lambda_{n}\left( \overset{\sim}{a} \right)} = {{{\sum\limits_{i = 0}^{{N/\eta} - 1}{\sum\limits_{l = 0}^{\eta - 1}{x_{{\eta {({n - i})}} + l}\left( {\prod\limits_{m = 1}^{K}a_{n - i - m}^{g_{ml}}} \right)}^{*}}}} - {{\sum\limits_{i = 0}^{{N/\eta} - 1}{\sum\limits_{l = 0}^{\eta - 1}{x_{{\eta {({n - i})}} + l}\left( {\prod\limits_{m = 1}^{K}a_{n - i - m}^{g_{ml}}} \right)}^{*}}}}}} & (38) \end{matrix}$

[0165] Likewise, using the (14) in the (15) we come to the (16), also due to the second embodiment of the present invention.

[0166] The case D) is referred to a modulation type characterized by pulses having different power, therefore the expression of the branch metric (35) can be used no more. The sole expression from which one shall start for the calculation of the branch metrics is the more general expression (30). In particular, the (17), (18), (19) and (21) continue to apply, we report for convenience:

α_(n)=μ_(n) p _(n)(17); c_(n)=μ_(n) q _(n)(18); q _(n) =p _(n) q _(n−1)  (19);

[0167] $\begin{matrix} {{{\overset{\sim}{q}}_{n}^{*}{\overset{\sim}{c}}_{n - i}} = {{{\overset{\sim}{\mu}}_{n - i}{\overset{\sim}{q}}_{n - i}{\overset{\sim}{q}}_{n}^{*}} = {{\overset{\sim}{\mu}}_{n - i}{\overset{\sim}{p}}_{n}^{*}{\prod\limits_{m = 1}^{i - 1}{\overset{\sim}{p}}_{n - m}^{*}}}}} & (21) \end{matrix}$

[0168] Branch metrics (30) can be expressed according to the information symbols in such a way that the Viterbi algorithm performs also the differential decoding. Multiplying to the purpose the terms ${{\sum\limits_{i = 0}^{N - 1}{x_{n - i}^{*}{\overset{\sim}{c}}_{n - i}}}}e{{\sum\limits_{i = 0}^{N - 1}{x_{n - i}^{*}{\overset{\sim}{c}}_{n - i}}}}$

[0169] of the (30) by |{tilde over (q)}_(n) ^(*)|=1, considering the (21) and the fact that the result is |{tilde over (c)}_(n)|=|{tilde over (α)}_(n)|=|{tilde over (μ)}_(n)|, we obtain: $\begin{matrix} {{\lambda_{n}\left( \overset{\sim}{a} \right)} = {{{\sum\limits_{i = 0}^{N - 1}{x_{n - i}^{*}{\overset{\sim}{\mu}}_{n - i}{\prod\limits_{m = 0}^{i - 1}{\overset{\sim}{p}}_{n - m}^{*}}}}} - {{\sum\limits_{i = 0}^{N - 1}{x_{n - i}^{*}{\overset{\sim}{\mu}}_{n - i}{\prod\limits_{m = 0}^{i - 1}{\overset{\sim}{p}}_{n - m}^{*}}}}} - {\frac{1}{2}{{\overset{\sim}{a}}_{n}}^{2}}}} & (39) \end{matrix}$

[0170] differing from the (22) relevant to the same case. On the basis of the (39) the state of the trellis diagram can be defined as in the (23), and identical results the number of states and the applicable techniques to reduce the complexity.

[0171] It is now removed the hypothesis of absence of intersymbol interference of the signal received, which means to consider the filter DISP of FIG. 1 present, which models the transmission channel, this time no more ideal. Consequently, also the possibility to consider modulated pulses having equal power subsides, and the complex envelope s(t, a) must be intended as coming out from the filter DISP.

[0172] In the case considered of coded linear modulation, the signal s(t, a) can also be expressed in the form: $\begin{matrix} {{s\quad \left( {t,a} \right)} = {\sum\limits_{n = 0}^{N_{T} - 1}{c_{n}{h\left( {t - {nT}} \right)}}}} & (0.3) \end{matrix}$

[0173] but now the impulse h(t) (known) considers also the dispersive filtering due to the transmission channel and therefore, does no more meet the Nyquist condition for the right reconstruction of the signal transmitted starting from its samples. Defining ${g_{n}\overset{\Delta}{=}{g({nT})}},$

[0174] where ${{g(t)}\overset{\Delta}{=}{{h(t)} \otimes {h^{*}\left( {- t} \right)}}},$

[0175] replacing the (0.3) in expression (24) of sequence ã estimated at the maximum likelihood by the non-coherent receiver, introducing the approximation logI₀(x){tilde over (=)}x, and proceeding as previously done to obtain the (25), we obtain the following decision strategy: $\begin{matrix} {\hat{a} = {\underset{\overset{\sim}{a}}{\arg \quad {mag}}{\left\{ {{{- \frac{1}{2}}{\sum\limits_{k,{= 0}}^{N_{T} - 1}{\sum\limits_{n = 0}^{N_{T} - 1}{{\overset{\sim}{c}}_{k}{\overset{\sim}{c}}_{n}^{*}g_{n - k}}}}} + {{\sum\limits_{n = 0}^{N_{T} - 1}{x_{n}{\overset{\sim}{c}}_{n}^{*}}}}} \right\}.}}} & (40) \end{matrix}$

[0176] As already made before, we indicated with {x_(n)} the sample sequence at the output of the matched filter FRIC (FIG. 2), which sequence represents a statistic sufficient to decide according to the (40) the symbols of the transmitted sequence. The samples {x_(n)} can be expressed as: $\begin{matrix} {x_{n} = {{\sum\limits_{l = {- L}}^{L}{g_{l}c_{n - l}^{j\theta}}} + n_{n}}} & (41) \end{matrix}$

[0177] having indicated with L the channel memory, being 2L+1 the number of non null samples of g(t), ${{e\quad n_{n}}\overset{\Delta}{=}{n({nT})}},$

[0178] where ${n(t)}\overset{\Delta}{=}{w\quad {(t) \otimes {{h^{*}\left( {- t} \right)}.}}}$

[0179] The complex discrete uncertain process n_(n) is gaussian of course, null mean and coloured, with autocorrelation function ${R_{a}(m)}\overset{\Delta}{=}{{E\quad \left\{ {n_{n}n_{n - m}^{*}} \right\}} = {2N_{0}{g_{m}.}}}$

[0180] An approach foreseeing the use of a matched filter in presence of ISI to obtain a sufficient statistic, though referred to coherent receivers, is described in the article by G. Ungerboeck, under the title <<ADAPTIVE MAXIMUM-LIKELIHOOD RECEIVER FOR CARRIER-MODULATED DATA-TRANSMISSION SYSTEM>>, published on IEEE Trans. Commun., vol. 22, pp. 624-635, May 1971.

[0181] An alternative procedure, and a preferred one as we will see, enabling to obtain a statistic sufficient to decode according to the (40), employing a whitened matched filter FRIC (WMF, Whitened Matched Filter) (FIG. 2) to filter the signal received, is described in the article by G. D. Forney Jr., under the title <<MAXIMUM-LIKELIHOOD SEQUENCE ESTIMATION OF DIGITAL SEQUENCES IN THE PRESENCE OF INTERSYMBOL INTERFERENCE>>, published on IEEE Inform. Theory, vol. 18, pp. 363-378, May 1972. It is understood that the teachings of the two above mentioned approaches do not injure the conception idea expressed in the (30), but must be considered complementary tothe same in the case ISI is present.

[0182] As it is known, a whitening filter is a filter such that the noise at its output has a constant power spectral density. Its presence is justified by the fact that, due to the ISI, the signal received is affected by coloured noise, while the expressions for the maximum likelihood sequence estimation were subject to the white noise assumption. The implementation of said filter is known to the skilled in the art, once the response to the impulse h(t) of the dispersive channel is known, and therefore of the impulse g(t). The whitened matched FRIC filter of FIG. 2 is in practice realized through the cascade of a filter matched with a whitening filter, in this case the sequence of samples z_(n) at output of such filter has been indicated with {z_(n)}. When the Forney approach shall be considered, these last shall replace the samples x_(n) in FIGS. 2 and 3.

[0183] The samples z_(n) can be expressed as:

z _(n) =y _(n) e ^(jθ) +w _(n)  (42)

[0184] where uncertain variables {w_(n)} are gaussian, null mean, independent and with variance σ_(w) ²=2N₀ and: $\begin{matrix} {y_{n}\overset{\Delta}{=}{\sum\limits_{l = 0}^{L}{f_{l}c_{n - l}}}} & (43) \end{matrix}$

[0185] where {ƒ_(n)} is the discrete time impulse response of the dispersion channel, obtained from the sequence {g_(n)} through the whitening filtering, mentioned before. It is possible to obtain in the known way an alternative formulation of the strategy of the non-coherent optimal receiver based on the sample sequence {z_(n)}, indicated hereinafter also with the vector z. To this purpose, starting from the density of probability p(z|ã,{tilde over (θ)}) and making the mean in respect to {tilde over (θ)}, we obtain the likelihood function p(z|ã) for the non-coherent best decision.

[0186] Using again the approximation logI₀(x){tilde over (=)}x, the decision strategy becomes: $\begin{matrix} {\hat{a} = {\underset{\overset{\sim}{a}}{\arg \quad \max}\left\{ {{{- \frac{1}{2}}{\sum\limits_{n = 0}^{N_{T} - 1}{{\overset{\sim}{y}}_{n}}^{2}}} + {{\sum\limits_{n = 0}^{N_{T} - 1}{z_{n}{\overset{\sim}{y}}_{n}^{*}}}}} \right\}}} & (44) \end{matrix}$

[0187] where {tilde over (y)}_(n) is defined according to {tilde over (c)}_(n) similarly to the (43).

[0188] We shall now describe the approaches that enable to obtain branch metrics concerning the two decision strategies (40) and (44), alternative between them. The intent is to use the Viterbi algorithm to evaluate the above mentioned expressions. The approaches foresee the introduction of appropriate approximations, in an almost similar way, as previously done starting from the (25). Out of the two approaches, the one referring to strategy (40) (Ungerboeck) shall be developed, a similar procedure enables to obtain the expression of the branch metric even in the more profitable case of the strategy (44) (Forney).

[0189] First of all, let's define the following sequence partial metric: $\begin{matrix} \begin{matrix} {{\Lambda_{n}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}\quad {{{\sum\limits_{k = 0}^{n - 1}{x_{k}{\overset{\sim}{c}}_{k}^{*}}}} - {\frac{1}{2}{\sum\limits_{k = 0}^{n - 1}{\sum\limits_{n = 0}^{n - 1}{{\overset{\sim}{c}}_{k}{\overset{\sim}{c}}_{n}^{*}g_{n - k}}}}}}} \\ {= \quad {{{\sum\limits_{k = 0}^{n - 1}{x_{k}{\overset{\sim}{c}}_{k}^{*}}}} - {\frac{1}{2}{\sum\limits_{k = 0}^{n - 1}\left\{ {{{{\overset{\sim}{c}}_{k}}^{2}g_{0}} + {2\quad {{Re}\quad\left\lbrack {\sum\limits_{l = 1}^{\min \quad {({L,k})}}{{\overset{\sim}{c}}_{k}{\overset{\sim}{c}}_{k - l}^{*}g_{l}^{*}}} \right\rbrack}}} \right\}}}}} \end{matrix} & (45) \end{matrix}$

[0190] where the property g_(n)=g_(−n) ^(*) has been used. After an initial transient, that is, for n≧L, we can define the following incremental metric: $\begin{matrix} \begin{matrix} {{\Delta_{n}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}\quad {{\Lambda_{n + 1}\left( \overset{\sim}{a} \right)} - {\Lambda_{n}\left( \overset{\sim}{a} \right)}}} \\ {= \quad {{{\sum\limits_{k = 0}^{n}{x_{k}{\overset{\sim}{c}}_{k}^{*}}}} - {{\sum\limits_{k = 0}^{n - 1}{x_{k}{\overset{\sim}{c}}_{k}^{*}}}} - {\frac{1}{2}{\left\{ {{{{\overset{\sim}{c}}_{n}}^{2}g_{0}} + {2\quad {{Re}\quad\left\lbrack {\sum\limits_{l = 1}^{L}{{\overset{\sim}{c}}_{n}{\overset{\sim}{c}}_{n - l}^{*}g_{l}^{*}}} \right\rbrack}}} \right\}.}}}} \end{matrix} & (46) \end{matrix}$

[0191] The sequence general metric Λ_(N) _(T) (ã) to maximize, can in this way be recurrently calculated. Due to the fact that the two summations in the (46) depend on the whole previous code sequence, the incremental metric has an unlimited memory that makes the calculation process difficult. Therefore, as in the case of ISI absence, the maximisation of the sequence general metric can be realized through a research on a tree diagram opportunely defined.

[0192] We can introduce now a truncation in the memory length of the incremental metric (46). To this purpose, in the first two summations of the (46) only the more recent N<<N_(T) terms are considered. The incremental metric obtained through this memory truncation is: $\begin{matrix} {{\lambda_{n}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}\quad {{{\sum\limits_{k = 0}^{N - 1}{x_{n - i}{\overset{\sim}{c}}_{n - i}^{*}}}} - \quad {{\sum\limits_{k = 0}^{N - 1}{x_{n - i}{\overset{\sim}{c}}_{n - i}^{*}}}} - {\frac{1}{2}\left\{ {{{{\overset{\sim}{c}}_{n}}^{2}g_{0}} + {2\quad {{Re}\quad\left\lbrack {\sum\limits_{l = 1}^{L}{{\overset{\sim}{c}}_{n}{\overset{\sim}{c}}_{n - l}^{*}g_{l}^{*}}} \right\rbrack}}} \right\}}}} & (47) \end{matrix}$

[0193] valid for n≧max{N−1,L}. As a consequence of the memory truncation, the maximization of the sequence metric can be now recurrently made through a research on a appropriately defined trellis diagram, using the Viterbi algorithm with branch metrics given by the (47). Also in this case the parameter N is comparable to a phase reconstruction memory.

[0194] Starting from the alternative approach (Forney) inborn in the strategy (44), using similar approximations a second non-coherent decoding diagram can be determined. The branch metrics obtained in this case can be expressed as follows: $\begin{matrix} {{\lambda_{n}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}{{{\sum\limits_{i = 0}^{N - 1}{z_{n - i}{\overset{\sim}{y}}_{n - i}^{*}}}} - {{\sum\limits_{i = 0}^{N - 1}{z_{n - i}{\overset{\sim}{y}}_{n - i}^{*}}}} - {\frac{1}{2}{{{\overset{\sim}{y}}_{n}}^{2}.}}}} & (48) \end{matrix}$

[0195] Even if the two non-coherent receivers based on branch metrics (47) and (48) derive from two equivalent formulations of the best non-coherent strategy, they do not give however the same performance, considering that the approximations introduced have not the same effect in the two cases. In the case of the strategy (40) (Ungerboeck approach), the sample x_(n) is correlated with the code symbol {tilde over (c)}_(n), while in the case of the strategy (44) (Forney approach), the sample z_(n) is correlated with {tilde over (y)}_(n). Along the corrected path, the sequences {z_(n)} and {{tilde over (y)}_(n)} differ only for the sequence of noise independent samples {w_(n)}, as it results from the (42), while sequences {x_(n)} and {{tilde over (c)}_(n)} are significantly different due to ISI and noise correlation. These two effects tend to cancel in the sequence general metric, while they are significant in the branch metrics (47). Therefore, determined the complexity of the receiver, the sub-optimal receiver based on the strategy (44) (Forney approach) gives significantly better performance and shall then be taken as reference for the development of the receiver valid for non linear modulations.

[0196] As for the implementation of a receiver, whose operation is based on the assumptions of the previous description, that takes into account the ISI, the receiver RIC of FIG. 2 continues to be valid, with the appropriate modifications in the METRIC(s) blocks, and provided that the correct FRIC filter is used, differently specified for the two implementation approaches.

[0197] Finally, it is considered the case of continuous phase non linear modulations, known with the acronym CPM (Continuous Phase Modulation), concerning the second embodiment of the present invention.

[0198] The complex envelope of a CPM signal, as known, has the form: $\begin{matrix} {{s\left( {t,a} \right)} = {\sqrt{\frac{2E_{s}}{T}}\exp \quad \left\{ {{j2}\quad \pi \quad h{\sum\limits_{n}{a_{n}q\quad \left( {t - {nT}} \right)}}} \right\}}} & (49) \end{matrix}$

[0199] where E_(s) is the power for information symbol, T is the symbol interval, h=k/p is the modulation index, (k and p are prime numbers between them), the information symbols {α_(n)} are assumed as independent, equipossible, and at values of the M-ary alphabet {±1,±3, . . . , ±(M−1)} and vector a indicates the sequence of information symbols. It is not definitely considered the case of use of channel coding techniques but the following considerations can be easily extended. The function q(t) is the phase response of the modulation and it is assumed that is satisfies the following normalization conditions: $\begin{matrix} {{q(t)} = \left\{ \begin{matrix} 0 & {{{per}\quad t} \leq 0} \\ {1/2} & {{{per}\quad t} \geq {L_{q}T}} \end{matrix} \right.} & (50) \end{matrix}$

[0200] where L_(q) is a positive integer and L_(q)T is the duration of the frequency pulse g(t) defined by: $\begin{matrix} {{g(t)}\overset{\Delta}{=}{\frac{{q(t)}}{t}.}} & (51) \end{matrix}$

[0201] Using a representation described in the following articles to express the (49):

[0202] P. A. Laurent, <<EXACT AND APPROXIMATE CONSTRUCTION OF DIGITAL PHASE MODULATIONS BY SUPERPOSITION OF AMPLITUDE MODULATED PULSES (AMP)>>, published on IEEE Trans. Commun., vol. 34, pp. 150-160, February 1986; e

[0203] U. Mengali e M. Morelli, <<DECOMPOSITION OF M-ary CPM SIGNALS INTO PAM WAVEFORMS>>, published on IEEE Trans. Information Theory, vol. 41, pp. 1265-1275, September 1995,

[0204] we come to the following correct expression of the complex envelope (49): $\begin{matrix} {{s\left( {t,a} \right)} = {\sum\limits_{k = 0}^{{Q^{\log_{2}M}{({M - 1})}} - 1}{\sum\limits_{n}{\alpha_{k,n}{h_{k}\left( {t - {nT}} \right)}}}}} & (52) \end{matrix}$

[0205] where M, for notation simplicity, is assumed to be a power of 2, ${Q\overset{\Delta}{=}2^{L_{q} - 1}},$

[0206] and the expressions of pulses {h_(k)(t)} and of symbols {α_(k,n)} as function of the sequence of the information symbols {a_(n)} are reported in the just mentioned paper of Mengali and Morelli. Cutting the summation (52) at the first K<Q^(log) ^(₂) ^(M) (M−1) terms, we obtain an approximation of s(t,a). As shown in the last mentioned article, the signal power is concentrated in the first M−1 components, that is, those associated to the pulses {h_(k)(t)} with 0<k≦M−2, called main pulses. Consequently, a value K=M−1 can be used in the (52) to obtain the best compromise between the quality of the approximation and the number of component signals, it has been demonstrated that a receiver based on the sole main pulses gives a performance practically coinciding with that of a best coherent receiver. For K=M−1, the approximation based on the sole main pulses can be lightly improved modifying the pulses {h_(k)(t)} in order to minimize the mean quadrant error between the signal and its approximation. For instance, for a quaternary CPM modulation, assuming K=3 and breaking off the information symbol α_(n) ∈{±1, ±3} in two binary symbols γ_(n,0) e γ_(n,1) belonging to the alphabet {±1} we can write:

α_(n)=2γ_(n,1)+γ_(n,0)  (53)

[0207] In the assumption that modulation indexes h and 2 h of the (49) are not integer, the symbols associated to the first three component signals can be expressed in the form: $\begin{matrix} \begin{matrix} {\alpha_{0,n} = {\alpha_{0,{n - 1}}^{j\quad h\quad \pi \quad a_{n}}}} \\ {\alpha_{1,n} = {\alpha_{0,{n - 1}}^{j\quad h\quad {\pi\gamma}_{n,1}}}} \\ {\alpha_{2,n} = {\alpha_{0,{n - 1}}^{j\quad h\quad {\pi\gamma}_{n,0}}}} \end{matrix} & (54) \end{matrix}$

[0208] With reference to FIG. 8, let's now introduce the sub-best non-coherent receiver for CPM signals, implemented as indicated by the reception process forming the object of the second embodiment of the present invention. As it can be noticed, the receiver RIC of FIG. 8 differs from that of FIG. 2 mainly due to the front-end structure, which in FIG. 2 consists of a sole reception filter RIC followed by the sampler CAMP, while in FIG. 8 it consists of a whitened matched multidimensional filter WMF (Whitened Matched Filter). The front-end WMF includes a bank of numeric filters matched to relevant pulses {h_(k)(t)}, supplied in parallel by the reception signal r(t), each one followed by a sampler at symbol cadence that sends its own samples {x_(k,n)} to the input of a unique whitening filter WF (Whitening Filter) of the multidimensional type from which the samples z_(n) come out. The remaining structure is identical in the two figures, pointing out that, in FIG. 8, the delay chain T is realized in a vectorial way, as well as vectorial is the calculation of the branch metrics made by the METRICTOT block, whose METRIC(s) blocks (FIG. 3) using (of course) different expressions versus those under the same name of FIG. 3.

[0209] Coming back to the procedure, we can easily demonstrate that the signals coming out from the filter bank matched to the pulses h_(k)(t) of FIG. 8, sampled at symbol frequency, represent a statistic sufficient to the non-coherent detection of a CPM signal. This assumed, it is convenient to have recourse to a simplified representation of a CPM signal, based on the sole main pulses, enabling to obtain a significant complexity reduction, in practice without any performance reduction.

[0210] The signal coming out from a filter matched to the pulse h_(k)(t), sampled at the instant nT, can be expressed in the form: $\begin{matrix} {\left( {{x_{k,n}\overset{\Delta}{=}{{r(t)} \otimes {h_{k}\left( {- t} \right)}}}} \right)_{t = {nT}} = {{s_{k,n}^{j\theta}} + n_{k,n}}} & (55) \end{matrix}$

[0211] where: $\begin{matrix} \left( {{n_{k,n}\overset{\Delta}{=}{{w(t)} \otimes {h_{k}\left( {- t} \right)}}}} \right)_{t = {nT}} & (56) \\ {\left( {{s_{k,n}\overset{\Delta}{=}{{s\left( {t,a} \right)} \otimes {h_{k}\left( {- t} \right)}}}} \right)_{t = {nT}} = {\sum\limits_{m = 0}^{K - 1}{\sum\limits_{i}{\alpha_{m,i}{g_{m,k}\left\lbrack {\left( {n - i} \right)T} \right\rbrack}}}}} & (57) \\ {{g_{m,k}(t)}\overset{\Delta}{=}{{h_{m}(t)} \otimes {{h_{k}\left( {- t} \right)}.}}} & (58) \end{matrix}$

[0212] e:

[0213] As it can be noticed from the (57), s_(k,n)≠α_(k,n) due to the ISI and the interference of the other component signals. Noise terms are also characterized by the mixed correlation function:

E{n _(m)(t)n _(k) ^(*)(t−τ)}32 2N₀g_(m,k)(−τ)  (59)

[0214] depending on the form of pulses g_(m,k)(t).

[0215] The whitening filter WF has been introduced in the front-end WMF of FIG. 8 since we found convenient to submit the sequences coming out from the different filters matched to a whitening filter of the type described above for linear modulated signals (Forney approach), it is possible in this way to extent to the CPM modulation the advantage of a better operation performance though with a reduced complexity of the RIC receiver. To this purpose we define: $\begin{matrix} {x_{n}\overset{\Delta}{=}\left( {x_{0,n},x_{1,n},\ldots \quad,x_{{K - 1},n}} \right)^{T}} & (60) \\ {s_{n}\overset{\Delta}{=}\left( {s_{0,n},s_{1,n},\ldots \quad,s_{{K - 1},n}} \right)^{T}} & (61) \\ {n_{n}\overset{\Delta}{=}\left( {n_{0,n},n_{1,n},\ldots \quad,n_{{K - 1},n}} \right)^{T}} & (62) \\ {{\overset{\_}{\alpha}}_{n} = \left( {\alpha_{0,n},\alpha_{1,n},\ldots \quad,\alpha_{{K - 1},n}} \right)^{T}} & (63) \\ {{G_{n}\overset{\Delta}{=}{\left\lbrack {g_{i,j}({nT})}\quad \right\rbrack \quad i}},{j = 0},1,\ldots \quad,{K - 1}} & (64) \end{matrix}$

[0216] representing in the order indicated, in matrix notation and at the discrete time instant n, sampled signals (x_(n)) coming out from the bank of K matched filters included in the WMF block, their signal (s_(n)) and noise (n_(n)) components, symbols {tilde over (α)}_(n) of the K component signals of Laurent representation (52), and the samples of the response to the pulse at output of the matched filter bank, grouped in a matrix (G_(n)) of K×K elements. By this matrix notation the vector of the observable, of K elements, can be expressed in the form: $\begin{matrix} {x_{n} = {{{s_{n}^{j\theta}} + n_{n}} = {{e^{j\theta}{\sum\limits_{l = {- L}}^{L}{G_{l}^{T}{\overset{\_}{\alpha}}_{n - l}}}} + n_{n}}}} & (65) \end{matrix}$

[0217] where L, tied to the duration L_(q)T of the frequency pulse, is a parameter representing the memory associated to the modulation process. The matrix covariance function of the vectorial process at noise discrete time n_(n) can also be defined as: $\begin{matrix} {{R_{n}(m)}\overset{\Delta}{=}{{E\left\{ {n_{n}n_{n - m}^{*T}} \right\}} = {{2N_{0}G_{- m}} = {2N_{0}G_{m}^{T}}}}} & (66) \end{matrix}$

[0218] where the property g_(m,k)(t)=g_(k,m) (−t) has been used.

[0219] Proceeding to the bilateral Z transform z [·] of matrix sequences previously introduced, we can define: $\begin{matrix} {{X(z)}\overset{\Delta}{=}{{z\left\lbrack x_{n} \right\rbrack} = {\sum\limits_{m = {- \infty}}^{\infty}{x_{m}z^{- m}}}}} & (67) \\ {{N(z)}\overset{\Delta}{=}{{z\left\lbrack n_{n} \right\rbrack} = {\sum\limits_{m = {- \infty}}^{\infty}{n_{m}z^{- m}}}}} & (68) \\ {{A(z)}\overset{\Delta}{=}{{z\left\lbrack {\overset{\rightarrow}{\alpha}}_{n} \right\rbrack} = {\sum\limits_{m = {- \infty}}^{\infty}{{\overset{\rightarrow}{\alpha}}_{m}z^{- m}}}}} & (69) \\ {{G(z)}\overset{\Delta}{=}{{z\left\lbrack G_{n} \right\rbrack} = {\sum\limits_{m = {- \infty}}^{\infty}{G_{m}z^{- m}}}}} & (70) \\ {{\overset{\rightarrow}{\Phi}(z)}\overset{\Delta}{=}{{z\left\lbrack {R_{n}(n)} \right\rbrack} = {{\sum\limits_{m = {- \infty}}^{\infty}{{R_{n}(m)}z^{- m}}} = {{2N_{0}{G\left( z^{- 1} \right)}} = {2N_{0}{{G^{T}(z)}.}}}}}} & (71) \end{matrix}$

[0220] The spectral matrix {overscore (Φ)}(z) of the vectorial process n_(n) is certainly defined not negative on the unit radius circumference. We shall assume that it is defined positive for the reasons described below. In fact if the determinant |{overscore (Φ)}(z)| were null as well on the circumference of unit radius, it would be simple to demonstrate that in this case the time-discrete uncertain processes {n_(k,n)} would be linearly dependent and therefore it should be possible to obtain an alternative sufficient statistic simply eliminating the outputs {x_(k,n)} whose noise components can be expressed, with unitary probability, as linear combination of the other ones. Strictly speaking, for CPM signals there is no reasonable probability that this possibility is verified. However, in some practical cases it can occur that the matrix {overscore (Φ)}(z) is inappropriately conditioned. In this case a simple countermeasure consists in eliminating some component signals. For instance, in the case of the quaternary CPM with raised cosine frequency pulse (RC, Rised Cosine) with L_(q)=2 (2RC), 2 main pulses, h₁(t) e h₂(t), are very similar. These pulses can be replaced by a mean pulse h_(e)(t) to which the symbol $\alpha_{e,k}\overset{\Delta}{=}{\alpha_{1,k} + {\alpha_{2,k}.}}$

[0221] corresponds.

[0222] In the hypothesis of spectral matrix {overscore (Φ)}(z) defined positive on the circumference of a unit radius it is possible to come to a factorization of the same that enables to perform the whitening filtering to obtain the advantages mentioned before. The factorization method can be obtained from the following articles:

[0223] <<THE FACTORIZATION OF DISCRETE-PROCESS SPECTRAL MATRICES>>, by P. R. Mothyka and J. A. Cadzow, published on IEEE Trans. Automat. Contr., vol. 12, pp. 698-707, December 1967;

[0224] <<FACTORIZATION OF DISCRETE-PROCESS SPECTRAL MATRICES>>, by D. N. Prabhakar Murthy, published on IEEE Trans. Inform. Theory, vol. 19, pp. 693-696, September 1973.

[0225] Applying the teaching contained in the following articles it is possible to find a matrix F(z) such that:

{overscore (Φ)}(z)=2N ₀ G ^(T)(z)=2N ₀ F(z ⁻¹)F ^(T)(z)  (72)

[0226] and such that the determinant |F(z⁻¹)| has no zeros inside the unit circle of convergence of the zed transform. Therefore a sufficient statistic, alternative to the one that can be obtained from a possible application of the Ungerboeck approach to the CPM modulation case, can be obtained filtering the vectorial signal {x_(n)} with a multidimensional filter WF of K×K elements whose transfer function is F⁻¹(z⁻¹). The vector resulting from the filtering is:

z _(n) =y _(n) e ^(jθ) +w _(n)  (73)

[0227] where {y_(n)} is the result of the total filtering suffered by the useful component of said modulated signal {s_(n)}.

[0228] The zed transform of the (65) is:

X(z)=G ^(T)(z)A(z)e ^(jθ) +N(z)=F(z ⁻¹)F ^(T)(z)A(z)e ^(jθ) +N(z).  (74)

[0229] Consequently, the zed transform of {y_(n)} is: $\begin{matrix} {{Y(z)} = {{\sum\limits_{m = {- \infty}}^{\infty}{y_{m}z^{- m}}} = {{F^{T}(z)}{{A(z)}.}}}} & (75) \end{matrix}$

[0230] We can demonstrate that the inverse transform of F(z) has only L+1 non-null elements F₁. Therefore the signal {y_(n)} can be expressed as: $\begin{matrix} {y_{n} = {\sum\limits_{l = 0}^{L}{F_{l}^{T}{\alpha_{n - l}.}}}} & (76) \end{matrix}$

[0231] As it can be noticed, the vectorial form of the (76) is similar to the one of the scalar expression (43) for linear modulation with ISI, irrespective of the coding, this result is a consequence of the development of Laurent of CPM signals.

[0232] Since the spectral matrix of the time-discrete process of noise w_(n) is:

{overscore (Φ)}_(w)(z)=F ⁻¹(z ⁻¹){overscore (Φ)}_(n)(z)F ^(−1T)(z)=2N ₀ I  (77)

[0233] the filter F⁻¹(z⁻¹) is a multidimensional whitening filter, coinciding with the WF filter of FIG. 3, which can be construed as a generalization of the whitening filter used in the case of linear modulations with ISI (Forney approach). For the physical implementation of such a filter it is necessary to introduce a delay in order to assure causality.

[0234] The WMF front-end of the RIC receiver of FIG. 8 can be construed as a multidimensional whitened matched filter with 1 input and K outputs, implemented through the cascade of a matched filter with 1 input and K outputs and of a whitening filter having size K×K. We have not considered the case of spectral matrix determinant {overscore (Φ)}_(n)(z) with zeros on the circumference of unit radius considering that it is not a case of practical importance in CPM modulations. However, this situation can be faced availing of the concept pole-zero deletion, also used by Forney to define the whitened matched filter in case of signals with zeroes in the band.

[0235] Using {z_(n)} as sufficient statistic, we can easily determine the best non-coherent decision strategy for CPM modulations. Continuing as indicated, to come to the strategy (44) (Forney approach), we can see that this strategy for CPM modulations is an extension of the (44) where a summation on the K components of the CPM signal is additionally present. Using the approximations already highlighted, which lead to the expression of the branch metric (48), branch metrics for CPM modulations assume now the following expression: $\begin{matrix} {{\lambda_{n}\left( \overset{\sim}{a} \right)}\overset{\Delta}{=}{{{\sum\limits_{k = 0}^{K - 1}{\sum\limits_{i = 0}^{N - 1}{z_{k,{n - i}}{\overset{\sim}{y}}_{k,{n - i}}^{*}}}}} - {{\sum\limits_{k = 0}^{K - 1}{\sum\limits_{i = 0}^{N - 1}{z_{k,{n - i}}{\overset{\sim}{y}}_{k,{n - i}}^{*}}}}} - {\frac{1}{2}{\sum\limits_{k = 0}^{K - 1}{{\overset{\sim}{y}}_{k,n}}^{2}}}}} & (78) \end{matrix}$

[0236] where {tilde over (y)}_(k,n) is defined in an obvious manner according to the (76) according to the hypothetical sequenceof information symbols. The number of states depends on N. For instance, using only the main pulses, that is a value K=M−1, for which {overscore (α)}_(n) depends only on α_(n), see the (53) and (54), the number of states of the trellis is S=M^(N+L−1) and can anyway be reduced employing the known techniques. It has been verified that, as in the case of linear modulations in absence of ISI, even using small values of N in the (78), the receiver RIC of FIG. 8 has a performance very similar to that obtained with coherent receivers used for modulated CPM signals.

[0237] The digital hardware of the RIC receiver of FIGS. 1, 2, 3, and 8 for the cases considered A), B), C), D) and CPM, and possible receivers derived from the same, can be conveniently implemented through digital integrated circuits of the ASIC type (Application Specific Integrated Circuit). This implementation method can be preferred in respect to the use of a mathematical microprocessor in the cases where the receiver must reach high operation speeds. The high operation speed that can be obtained, can be drawn in itself from the modularity of the METRICTOT structure shown in FIG. 3, enabling the parallel processing of branch metrics.

[0238] Therefore, while particular embodiments of the present invention have been shown and described, it should be understood that the present invention is not limited thereto since other embodiments may be made by those skilled in the art without departing from the true spirit and scope thereof. It is thus contemplated that the present invention encompasses any and all such embodiments covered by the following claims. 

1. Non-coherent reception process of information symbol sequences, obtained by amplitude and/or phase digital modulation of a carrier, transmitted on a communication channel, affected by additive white gaussian noise, based on a particularly effective use of the Viterbi algorithm applied to a trellis sequential diagram, or trellis, whose branches represent all the possible transitions among states defined by all possible subsequences of information symbols, possibly encoded, having finite length, through which algorithm at each symbol interval, paths are selected on the trellis such that a cumulative path metric, or general metric or sequence metric, of transition metrics, or branch metrics, is maximum; said path metric being indicative of the likelihood degree existing among symbols of a path associated to the same path metric ({{tilde over (α)}_(n)},{{tilde over (c)}_(n)}) and a sequence of transmitted symbols ({α_(n)},{c_(n)}), characterized in that each said transition metric (λ_(n) ^((s))) is calculated through the following steps: a) non-coherent conversion in base band of the signal received, subsequent filtering of the converted signal (r(t)) through a filter (FRIC) matched to the transmitted pulse and symbol frequency sampling of the filtered signal, obtaining a sequence of complex samples ({x_(n)}); b) construction of a phase reference through accumulation of N−1 products among said complex sequential samples (x_(n−1), . . . ,x_(n−N+1)), conjugated, and corresponding code symbols ({tilde over (c)}_(n−1), . . . , {tilde over (c)}_(n−N+1)), also complex, univocally associated to a relevant said branch of the trellis; the number N−1 being the finite length, selected in order to obtain the desired accuracy in the constructed phase, said accuracy increasing as N increases, at the expense of possible increase in the trellis complexity, expressed in terms of number of states; c) normalization of the value of said phase reference, through division by the module itself; d) replacement of a phase reference, or phasor (e^(−jθ)) of said modulated carrier, present in the known analytical expression of transition metrics used by an optimal coherent receiver, which could replace said non coherent receiver (RIC) whenever said phasor would be known, with said phase reference resulting from said step c), obtaining an analytical expression for the calculation of each of said transition metric (λ_(n) ^((s))) used by said non coherent receiver(RIC).
 2. Non-coherent reception process according to claim 1 , characterized in that, in presence of linear modulations, said analytical expression for the calculation of transition metrics, at the n-th discrete instant, is the following: $\lambda_{n} = {\frac{{Re}\left\{ {\sum\limits_{i = 1}^{N - 1}{x_{n}x_{n - i}^{*}{\overset{\sim}{c}}_{n}^{*}{\overset{\sim}{c}}_{n - i}}} \right\}}{{\sum\limits_{i = 1}^{N - 1}{x_{n - i}^{*}{\overset{\sim}{c}}_{n - i}^{*}}}} - \frac{{{\overset{\sim}{c}}_{n}}^{2}}{2}}$

where: λ_(n) is a generic said branch metric; ({tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1)) are N code symbols univocally associated to a relevant branch of the trellis; (x_(n−1), . . . , x_(n−N+1) ) are N sequential samples obtained filtering said converted signal (r(t)) through said filter (FRIC) matched to the transmitted pulse; the asterisk (*)denoting conjugate complex values.
 3. Non coherent reception process according to claim 2 , characterized in that when said code symbols ({c_(n)}) correspond to M discrete phase values of said modulated carrier, or M-PSK, said analytical expression for the calculation of the transition metrics, at the n-th discrete instant, is the following: $\lambda_{n} = {{Re}\left\{ {\sum\limits_{i = 1}^{N - 1}{x_{n}x_{n - i}^{*}{\overset{\sim}{c}}_{n}^{*}{\overset{\sim}{c}}_{n - i}}} \right\}}$

where: λ_(n) is a generic said transition metric; {tilde over (c)}_(n−1) are N M-PSK symbols univocally associated to a relevant said branch of the trellis; and x⁻¹ are N sequential samples obtained filtering said converted signal (r(t)) through said filter (FRIC) matched to the transmitted pulse; the asterisk (*)denoting complex conjugate symbols and samples.
 4. Non coherent reception process according to claim 3 , characterized in that when the coding used is differential, said analytical expression for the calculation of the transition metrics, at the n-th discrete instant, is the following: $\lambda_{n} = {{Re}\left\{ {\sum\limits_{i = 1}^{N - 1}{x_{n}x_{n - i}^{*}{\prod\limits_{m = 0}^{i - 1}{\overset{\sim}{a}}_{n - m}^{*}}}} \right\}}$

where the ã_(n−m) ^(*) are N M-PSK symbols univocally associated to a relevant said branch of the trellis.
 5. Non coherent reception process according to claim 2 , characterized in that, when said modulation is implemented through M discrete phase and amplitude values, or M-QAM, and said coding is quadrant differential, said analytical expression for the calculation of the transition metrics, at the n-th discrete instant, is the following: $\lambda_{n} = {\frac{{Re}\left\{ {\sum\limits_{i = 1}^{N - 1}{x_{n}x_{n - i}^{*}{\overset{\sim}{a}}_{n}^{*}{\overset{\sim}{\mu}}_{n - i}{\prod\limits_{m = 1}^{i - 1}{\overset{\sim}{p}}_{n - m}^{*}}}} \right\}}{{\sum\limits_{i = 1}^{N - 1}{x_{n - i}^{*}{\overset{\sim}{\mu}}_{n - i}{\overset{\sim}{p}}_{n}^{*}{\prod\limits_{m = 1}^{i - 1}{\overset{\sim}{p}}_{n - m}^{*}}}}} - \frac{{{\overset{\sim}{a}}_{n}}^{2}}{2}}$

where: λ_(n) is a generic said transition metric; {tilde over (α)}_(n) are N M-QAM symboisunivocally associated to a relevant said branch of the trellis; {tilde over (p)}_(n) are N symbols that assume values {±1, ±j}; {tilde over (μ)}_(n) is the symbol ã_(n) multiplied by a phasor performing a rotation of an angle multiple of π/2 that brings it in the first quadrant of the complex plane; and x_(n) are N sequential samples obtained filtering said converted signal (r(t)) though said filter (FRIC) matched to the transmitted pulse; the asterisk (*) denoting complex conjugate symbols and samples.
 6. Non coherent reception process according to claim 2 , characterized in that when said modulation is implemented through M discrete phase values, or M-PSK, and said coding is convolutional, said analytical expression for the calculation of the transition metrics, at the n-th discrete instant, is the following: $\lambda_{n} = {{Re}\left\{ {\sum\limits_{i = 1}^{L - 1}{\sum\limits_{j = 0}^{\eta - 1}{\sum\limits_{l = 0}^{\eta - 1}{x_{{\eta \quad n} + j}\left( {x_{{\eta {({n - i})}} + l}^{*}\left( {\prod\limits_{m = 1}^{K}{\overset{\sim}{a}}_{n - m}^{g_{mj}}} \right)} \right)^{*}{\prod\limits_{k = 1}^{K}{\overset{\sim}{a}}_{{({n - i})} - k}^{g_{kl}}}}}}} \right\}}$

where: λ_(n) is a generic said transition metric; {tilde over (α)}_(n) are L+K−1 M-PSK symbols univocally associated to a relevant said branch of the trellis; x_(n) are N sequential samples obtained filtering said converted signal (r(t)) though said filter (FRIC) matched to the transmitted pulse; the asterisk (*) denoting complex conjugate symbols and samples; K is the code constraint length; and finally η is a number of K-uple g_(mj), g_(kl) of constants defining the code generators, such that the code rate is 1/η.
 7. Non coherent reception process of information symbol sequences obtained through phase and/or amplitude digital modulation of a carrier, transmitted on a communication channel affected by additive white gaussian noise, based on a particularly effective use of the Viterbi algorithm applied to a sequence trellis diagram, or trellis, whose branches represent all the possible transitions among states defined by all possible sub-sequences of information symbols, possibly encoded, of finite length, through which algorithm, at each symbol interval, paths are selected on the trellis such that a cumulative path metric, of transition metrics, or branch metrics, is maximum, said path metric being indicative of the likelihood degree existing among the symbols of the path associated to the same path metric ({{tilde over (α)}_(n)},{{tilde over (c)}_(n)}) and a sequence of transmitted symbols $\frac{i}{\eta},$

characterized in that each said transition metric (λ_(n) ^((s))) is calculated through the following steps: a) non coherent conversion in base band of the signal received, subsequent filtering of the converted signal (r(t)) through a filter (FRIC) matched to the transmitted pulse and sampling at symbol frequency of the filtered signal, obtaining a sequence of complex samples ({x_(n)}); b) identification of a function to maximize equal to the known expression for maximum likelihood sequence estimation of a non-coherent receiver, as expression of a general metric associated to said complex sample sequence ({x_(n)}); c) expression of a partial metric, obtained considering said general metric up to a current n-thsample of said sequence of complex samples ({x_(n)}); d) expression of an incremental metric, of unlimited memory, obtained from the difference between the expression of said partial metric at a current signalling interval and at an immediately preceding interval; e) truncation of the length of said unlimited memory at N−1 samples of said sequence {x_(n)} preceding the current sample, obtaining the analytical expression of a said branch metric (λ_(n) ^((s))) of said trellis diagram, built on the basis of all the possible symbol subsequences ({{tilde over (α)}_(n)},{{tilde over (c)}_(n)}) having length N; f) calculation of said general metric through recurrent updating of said sequence partial metric.
 8. Non coherent reception process according to claim 7 , characterized in that in the presence of linear modulations, said analytical expression for the calculation of the branch metrics (λ_(n) ^((s))), at the n-th discrete instant, is the following: $\lambda_{n} = {{{\sum\limits_{i = 0}^{N - 1}{x_{n - i}{\overset{\sim}{c}}_{n - i}^{*}}}} - {{\sum\limits_{i = 0}^{N - 1}{x_{n - i}{\overset{\sim}{c}}_{n - i}^{*}}}} - {\frac{1}{2}{{{\overset{\sim}{c}}_{n}}^{2}.}}}$

where: λ_(n) is a generic said branch metric; {tilde over (c)}_(n−1) are N code symbols univocally associated to a relevant said branch of the trellis; x_(n−1) are N sequential samples obtained filtering said converted signal (r(t)) through said filter (FRIC) matched to the transmitted pulse; the asterisk (*) denoting complex conjugate values.
 9. Non coherent reception process according to claim 8 , characterized in that in the presence of modulated pulses of equal power, or M-PSK, said analytical expression for the calculation of the branch metrics (λ_(n) ^((s))), at the n-th discrete instant; is simplified as follows: $\lambda_{n} = {{{\sum\limits_{i = 0}^{N - 1}{x_{n - i}{\overset{\sim}{c}}_{n - i}^{*}}}} - {{{\sum\limits_{i = 0}^{N - 1}{x_{n - i}{\overset{\sim}{c}}_{n - i}^{*}}}}.}}$


10. Non coherent reception process according to claim 8 , characterized in that when said modulation is followed by a differential encoding according to c_(n)=c_(n−1)α_(n), said analytical expression for the calculation of branch metrics (λ_(n) ^((s))), at the n-th discrete instant, is the following: $\lambda_{n} = {{{\sum\limits_{i = 0}^{N - 1}{x_{n - i}{\prod\limits_{m = 0}^{i - 1}{\overset{\sim}{a}}_{n - m}^{*}}}}} - {{\sum\limits_{i = 0}^{N - 1}{x_{n - i}{\prod\limits_{m = 0}^{i - 1}{\overset{\sim}{a}}_{n - m}^{*}}}}}}$

where the ã_(n−m) ^(*) are N M-PSK symboisunivocally associated to a relevant said branch of the trellis.
 11. Non coherent reception process according to claim 8 , characterized in that when said modulation is implemented through M discrete values of phase and amplitude, or M-QAM, and said coding is quadrant differential, said analytical expression for the calculation of branch metrics (λ_(n) ^((s))), at the n-th discrete instant, is the following: $\lambda_{n} = \left. {{{\sum\limits_{i = 0}^{N - 1}{x_{n - i}^{*}{\overset{\sim}{\mu}}_{n - i}{\prod\limits_{m = 0}^{i - 1}\quad {\overset{\sim}{p}}_{n - m}^{*}}}}} - {{\sum\limits_{i = 1}^{N - 1}{x_{n - i}^{*}{\overset{\sim}{\mu}}_{n - i}{\prod\limits_{m = 0}^{i - 1}\quad {\overset{\sim}{p}}_{n - m}^{*}}}}} - \frac{1}{2}} \middle| {\overset{\sim}{a}}_{n} \right|^{2}$

where: ã_(n) are N M-QAM symbols univocally associated to a relevant said branch of the trellis; {tilde over (p)}_(n) are N symbols assuming values {±1, ±j}; {tilde over (μ)}_(n) is the symbol {tilde over (α)}_(n) multiplied by a phasor performing a rotation of an angle multiple of π/2 that brings it in the first quadrant of the complex plane.
 12. Non coherent reception process according to claim 8 , characterized in that, when said modulation is implemented through M discrete phase values, or M-PSK, and said coding is convolutional, said analytical expression for the calculation of branch metrics (λ_(n) ^((s))), at the n-th discrete instant, is the following: $\lambda_{n} = {{{\sum\limits_{i = 0}^{{N/\eta} - 1}{\sum\limits_{l = 0}^{\eta - 1}{x_{{\eta {({n - i})}} + l}{\overset{\sim}{c}}_{{\eta {({n - i})}} + l}^{*}}}}} - {{\sum\limits_{i = 0}^{{N/\eta} - 1}{\sum\limits_{l = 0}^{\eta - 1}{x_{{\eta {({n - i})}} + l}{\overset{\sim}{c}}_{{\eta {({n - i})}} + l}^{*}}}}}}$

where: indexes n and i run over the information symbols, l scans the code symbols associated to the (n−i)-th information symbol, and 1/η is the code rate.
 13. Non coherent reception process according to claim 7 , characterized in that in the presence of a linear modulation and a dispersive channel the signal converted in base band (r(t)), filtered by said filter (FRIC) matched to the transmitted pulse, is additionally filtered by a whitening filter (WF) such that a coloured noise at the input of said filter yelds at the output of the same a spectral density of constant power.
 14. Non coherent reception process according to claim 13 , characterized in that complex samples coming out from said whitening filter assume the following expression: z_(n)=y_(n)e^(jθ)+w_(n); where $y_{n} = {\sum\limits_{l = 0}^{L}{f_{l}c_{n - l}}}$

being {ƒ_(n)} the discrete time pulse response of the dispersive channel, and {w_(n)} a sequence of random pulses reprresenting said whitened noise.
 15. Non coherent reception process according to claim 14 , characterized in that said analytical expression for ihe calculation of branch metrics (λ_(n) ^((s))), at the n-th discrete instant, is the following: $\lambda_{n} = {{{\sum\limits_{i = 0}^{N - 1}{z_{n - i}{\overset{\sim}{y}}_{n - i}^{*}}}} - {{\sum\limits_{i = 1}^{N - 1}{z_{n - i}{\overset{\sim}{y}}_{n - i}^{*}}}} - {\frac{1}{2}{{{\overset{\sim}{y}}_{n}}^{2}.}}}$


16. Non coherent reception process according to claim 7 , characterized in that in the presence of non linear modulations of the continuous phase type, otherwise called CPM (Continuous Phase Modulations), such that the modulated signal s(t,a) can be expressed as summation of K pulses h_(k)(t) multiplied by relevant symbols (α_(k,n)), said signal converted in base band (r(t)) is filtered by a bank of K filters placed in parallel, each one matched to a relevant pulse of the summation, and the K numeric samples of the filtered signals, grouped in an algebraic vector x_(n) of K elements, are additionally filtered by a multidimensional whitening filter (WF), organized in the form of matrix of K×K elements, such that a coloured noise vector n_(n) at the input of said whitening filter yelds at the output a spectral density of constant power and uncorrelated components.
 17. Non coherent reception process according to claim 16 , characterized in that the complex samples coming out from said whitening filter form a vector z_(n) that assumes the following expression: z_(n)=y_(n)e^(jθ)+w_(n) where {y_(n)} is the result of the global filtering underwent by the useful component of said modulated signal {s_(n)}, and w_(n) is a vector whose elements are obtained from the whitening filtering of corresponding elements of said noise vector n_(n).
 18. Non coherent reception process according to claim 17 , characterized in that said analytical expression for the calculation of branch metrics (λ_(n) ^((s))), at the n-th discrete instant, is the following: $\lambda_{n} = {{{\sum\limits_{k = 0}^{K - 1}{\sum\limits_{i = 0}^{N - 1}{z_{k,{n - i}}{\overset{\sim}{y}}_{k,{n - i}}^{*}}}}} - {{\sum\limits_{k = 0}^{K - 1}{\sum\limits_{i = 0}^{N - 1}{z_{k,{n - i}}{\overset{\sim}{y}}_{k,{n - i}}^{*}}}}} - {\frac{1}{2}{\sum\limits_{k = 0}^{K - 1}{{{\overset{\sim}{y}}_{k,n}}^{2}.}}}}$


19. Non coherent reception process according to any of the previous claims, characterized in that when said states are defined by a number of symbols less than N−1, the missing code symbols to the purposes of the calculation of said metrics being found in a relevant path surviving to said selection on the trellis operated by the Viterbi algorithm,.
 20. Non coherent reception process according to any of the previous claims, characterized in that when said sequences of information symbols include also pilot symbols, known at the reception side, and at a discrete k-th instant a said pilot symbol is recognized, said cumulative metrics are subsequently calculated only on paths ending in states compatible with said pilot symbol.
 21. Non coherent reception process according to any of the previous claims, characterized in that when said coded symbols ({c_(n)}), are information symbols ({α_(n)}) undergoing a differential coding (5) followed by a channel coding (6) invariant to phase rotations of said carder, said trellis is built on said information symbols ({α_(n)}), said Viterbi algorithm performing the maximum likelihood estimation of a sequence of said information symbols ({α_(n)}).
 22. Non coherent receiver of sequences of coded symbols ({c_(n)}) obtained by amplitude and/or phase digital modulation of a carrier, transmitted on a communication channel, affected by additive white gaussian noise, including: a non coherent converter in base band of the signal received, followed by a filter (FRIC) matched to the transmission pulse, followed in its turn by a sampler (CAMP) at symbol frequency, which obtains a sequence of complex samples {x_(n)}; a phase reconstruction memory (SHF1) in which N−1 samples (x_(n−1), . . . ,x_(n−N+1)) of said sequence {x_(n)} preceding a current sample (x_(n)) are stored; calculation means (METRICTOT) of transition metrics (λ_(n) ^((s))), or branch metrics, of a trellis sequential diagram, or trellis, whose branches represent all possible transitions among states defined by all possible subsequences of information symbols ({c_(n)}) of finite length; a Viterbi processor adapted to select paths on the trellis such that a cumulative path metric of transition metrics, or branch metrics, is maximum, said path metric indicating the likelihood level existing among symbols ({{tilde over (c)}_(k)}) of a relevant path and a sequence of transmitted symbols ({c_(n)}), characterized in that said calculation means (METRICTOT) of the transition metrics (λ_(n) ^((s))), are subdivided into a plurality of identical sub-means (METRIC(s)), each of them being adapted to calculate a relevant transition metric (λ_(n) ^((s))), including: a memory for N code symbols ({tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1)) univocally associated to a relevant branch of the trellis; construction means of a phase reference controlled by said N−1 samples (x_(n−1), . . . ,x_(n−N+1)) stored in said phase reconstruction memory (SHF1) and by N−1 said code symbols ({tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1)) corresponding to said samples; first product means (PROD) adapted to multiply said current sample (x_(n)) by the conjugate of a said code symbol ({tilde over (c)}_(n)) corresponding to said current sample; second product means (PROD) adapted to multiply the value coming out from said first product means by the conjugate of said reconstructed phase reference; means (REAL) adapted to extract the real part of the value coming out from said second product means; means (MOD) adapted to calculate the modulus of said constructed phase reference; normalization means (DIV) adapted to divide said real part by said modulus; first adder means (4) of the quotient coming out from said normalization means (DIV) with the square modulus, changed in sign and divided by two, of said code symbol ({tilde over (c)}_(n)) corresponding to said current sample (x_(n)), obtaining a said relevant transition metric (λ_(n) ^((s))).
 23. Receiver according to claim 22 , characterized in that said phase reference construction means include: N−1 third product means (PROD) adapted to multiply said N−1 samples (x⁻¹, . . . ,x_(n−N+1)) stored in said phase reconstruction memory (SHF1) by complex conjugate of said corresponding N−1 code symbols ({tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1)); and second adder means (3) adapted to add among them the N−1 values coming out from said third product means, constructing said phase reference; the number N−1 being said finite length, selected in order to obtain the desired accuracy in the constructed phase, said accuracy increasing as N increases.
 24. Non-coherent receiver of symbol sequence ({α_(n)}), obtained impressing M phase discrete values to a carrier transmitted on a communication channel, affected by white gaussian noise, including: a non-coherent converter in base band of the signal received, followed by a filter (FRIC) matched to the transmission pulse, followed in its turn by a sampler (CAMP) at symbol frequency, that obtains a sequence of complex samples {x_(n)}; a phase reconstruction memory (SHF1) where N−1 samples (x_(n−1), . . . ,x_(n−N+1)) of said sequence {x_(n)} preceding a current sample (x_(n)) are stored; calculation means (METRICTOT) of transition metrics, or branch metrics, (λ_(n) ^((s))) of a trellis sequential diagram, or trellis, whose branches represent all possible transitions among states defined by all possible subsequences of information symbols ({c_(n)}) of finite length; a Viterbi processor adapted to select paths on the trellis such that a cumulative path metric of transition metrics, or branch metrics, is maximum, said path metric indicating the likelihood degree existing among the symbols ({{tilde over (c)}_(n)}) of a relevant path and a transmitted sequence symbols ({c_(n)}), characterized in that said calculation means (METRICTOT) of the transition metrics (λ_(n) ^((s))), are subdivided into a plurality of identical sub-means (METRIC(s)), each of them calculating a relevant transition metric (λ_(n) ^((s))), including: a memory for N code symbols ({tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1)) univocally associated to a relevant branch of the trellis; phase reference construction means controlled by said N−1 samples (x_(n−1), . . . ,x_(n−N+1)) stored in said phase reconstruction memory (SHF1) and by N−1 said code symbols ({tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1)) corresponding to said samples; first product means (PROD) adapted to multiply said current sample (x_(n)) by the conjugate of a said code symbol ({tilde over (c)}_(n)) corresponding to said current sample; second product means (PROD) adapted to multiply the value coming out from said first product means by the conjugate of said constructed phase reference; means (REAL) adapted to extract the real part of the value coming out from said second product means, obtaining a said relevant transition metric (λ_(n) ^((s))).
 25. Non-coherent receiver according to claim 24 , characterized in that, said phase reference construction means include: N−1 third product means (PROD) adapted to multiply said N−1 samples (x_(n−1), . . . ,x_(n−N+1)) stored in said phase reconstruction memory (SHF1) by the complex conjugate of said corresponding N−1 code symbols ({tilde over (c)}_(n−1), . . . ,{tilde over (c)}_(n−N+1)); and adder means (3) adapted to add among them the N−1 values coming out from said third product means, and also adapted to construct said phase reference; the number N−1 being said finite length, selected in order to obtain the desired accuracy in the constructed phase, said accuracy increasing as N increases. 